Extended Essay- Mathematics - GitHub Pagesoscarw3.github.io/files/wang_oscar_EE_mathematics.pdf · Oscar Wang Chinese International School Candidate No.: 0637 095 Extended Essay-

Oscar Wang Chinese International School

Candidate No.: 0637 095

Extended Essay- Mathematics

What is the multinomial theorem, what is its relationship with combinatorics and how can it be applied to both real and

abstract fields?

Word count: 3418

Oscar Wang Proof and Applications of the Multinomial Theorem and its relationship with Combinatorics

ii

Abstract

In this extended essay, I look into what the multinomial theorem is, its relationship with combinatorics, as well as how it can be applied to both real and abstract fields.

I first prove the binomial theorem through the principle of mathematical induction, and by extending the proof for the binomial theorem, I predict the multinomial theorem and prove it, again through the principle of mathematical induction:

!

(a1 + a2 + a3 + ...am )n =

n!n1!n2!n3!n4!...nm!

a1n1a2

n2a3n 3 ...am

nm

n1 ,n2 ,n3 ...nm

"

I then apply the multinomial theorem to both applied and abstract fields of mathematics. I relate the multinomial coefficient with combinatorics, as well as Pascal’s triangle. This is then extended to Pascal’s hyper-pyramids, a multidimensional version of Pascal’s triangle. I investigate the trends of both the hyper-pyramids and Pascal’s triangle. I also discover that the multinomial theorem is applicable to poker dice, a real life application, and can be used to calculate the probability of certain results in the game.

The proof by mathematical induction shows that there is indeed a multinomial theorem that can be extended from the binomial theorem, and that the use of the multinomial theorem is applicable to many fields of mathematics. Whilst the theorem is currently applicable to hyper-pyramids and poker dice, if further research is done, the multinomial theorem could be applied to various other mathematical fields in the future, from game theory to cryptology. (217 words)


iii

Table Of Contents

1. Introduction ....................................................................................................... 1

1.1. Why did I choose to investigate the multinomial theorem? .......................... 1

1.2. Defining the multinomial theorem and related fields ................................... 1

2. Proof of the Multinomial Theorem..................................................................... 1

2.1. Proof of the binomial theorem through mathematical induction................... 2

2.2. Examples with expansion of different multinomials and conjecture............. 3

2.3. Proof of Multinomial Theorem with Mathematical Induction ...................... 5

2.3.1. Mechanism for Multinomial Theorem.................................................. 6

3. Multinomial Theorem with combinatorics.......................................................... 7

4. Multinomial Theorem and Combinatorics through geometry.............................. 8

4.1. Pascal’s Triangle......................................................................................... 8

4.2. Pascal’s Pyramid......................................................................................... 8

4.3. Pascal’s Hyper-Pyramids ...........................................................................11

4.4. Summation of row and layers with Pascal’s Hyper-Pyramids .....................11

5. Multinomial Theorem & Combinatorics with real life application (poker dice)..13

6. Conclusion ........................................................................................................15

7. Works Cited......................................................................................................16


1

1. Introduction

1.1. Why did I choose to investigate the multinomial theorem?

The binomial theorem was the first topic that I learnt in Mathematics Higher Level. A binomial is a polynomial with two terms, in the form a+b, and the binomial theorem is a theorem regarding the expansion of the binomials (Wolfram|Alpha: Math Reference: binomial). Binomials expressed in the form (a+b)n can be expanded with the following formula (Binomial theorem - Topics in precalculus):

!

(a + b)n = (nCr)r=0

n

" an#rbr

I was initially slightly overwhelmed by the proof of the theorem when it was just introduced, as I had just learned both the theorem and mathematical induction; however, as I began to understand the topic, I became more and more intrigued. The binomial theorem has implications in probability, combinatorics and geometry, which can be explained by different proofs through mathematical induction. The elegance and beauty of the way the binomial theorem is able to relate to so many different aspects of mathematics astounded me. As a result, I hope to develop and expand my knowledge of the topic through the multinomial theorem, and experiment with how the theorem could be applied to both practical and abstract mathematical fields. In my extended essay, I will answer the question “What is the multinomial theorem, what is its relationship with combinatorics and how can it be applied to both real and abstract fields?” I would also like to note that even though most of the proofs can be found online, all proofs in this essay were done independently, unless otherwise stated.

1.2. Defining the multinomial theorem and related fields

The multinomial theorem is a corollary of the binomial theorem (Bona 70). Whereas the binomial theorem is a formula for the expansion of a polynomial with two terms to the nth power, the multinomial is a formula for the expansion of a multinomial with m terms to the nth power (Multinomial Series -- from Wolfram MathWorld). As a result, the multinomial theorem is an extension of the binomial theorem. Combinatorics is the study of mathematics that allows us to count and determine the number of possible outcomes (Combinatorics -- from Wolfram MathWorld). Combinatorics relates to both the permutations and combinations of different sets. Whilst permutations are focused on different possible rearrangements of a certain number of objects, combinations are focused on different possible subsets of objects from a larger set of objects. In other words, permutations take into consideration the order of the objects, whilst combinations do not. Since the binomial coefficients are essentially different combinations, both the binomial theorem and multinomial theorem is ultimately linked to combinatorics and combinations specifically. This relation between the coefficients and combinatorics leads to geometric patterns and sequences in the form of Pascal’s Triangle and Pascal’s Hyper-Pyramids (Horn 1-2).

2. Proof of the Multinomial Theorem


2

2.1. Proof of the binomial theorem through mathematical induction.

The proof of the binomial theorem by the principle of mathematical induction is a proof explored in the IB Mathematics Higher Level class, and can be done simply with the level of induction learnt in class. Though rather simple, it is essential for the understanding and proof of the Multinomial Theorem.

Let S(n) be the statement:

!

(a + b)n = (nCr)r=0

n

" an#rbr

for all

!

n"Z +

for

!

n =1:

!

LHS = (a + b)1 = a + b

RHS = (nCr)r=0

1

" a1#rbr=1C0a1#0b0+1C1a

1#1b1=a1+b1= a + b

LHS = RHS

S(1) is true. Lets assume S(n) is true for some value k, that is:

!

(a + b)k = (kCr)r=0

k

" ak#rbr

for all

!

k "Z + for

!

n = k +1:

!

LHS = (a + b)k+1

= (a + b)k (a + b)1

= ( (kCr)r= 0

k

" ak#rbr )(a + b)

=(kC0akb0+kC1a

k#1b1+kC2ak#2b2 ...+kCk#2a

2bk#2+kCk#1a1bk#1+kCka

0bk )(a + b)= a(kC0a

kb0+kC1ak#1b1+kC2a

k#2b2 ...+kCk#2a2bk#2+kCk#1a

1bk#1+kCka0bk ) +

b(kC0akb0+kC1a


2bk#2+kCk#1a1bk#1+kCka

0bk )=(kC0a

k+1b0+kC1akb1+kC2a

k#1b2 ...+kCk#2a3bk#2+kCk#1a

2bk#1+kCka1bk ) +

(kC0akb1+kC1a


2bk#1+kCk#1a1bk+kCka

0bk+1)

Add all the variables with the same exponents, that is

!

axby+axby

!

LHS = ak+1b0+(kC1+kC0)akb1+(kC2+kC1)a

k"1b2+(kC3+kC2)ak"2b3 ... +(kCk"1+kCk"2)a

2bk"1+(kCk+kCk"1)a1bk+a0bk+1

Since the yth term from the 1st part of the equation is added to the (y-1)th term from the 2nd part of the equation,

!

(kCy) +(kCy"1) is the coefficient for the variable

!

axby


3

!

(kCy) +(kCy"1) =k!

y!(k " y)!+

k!(y "1)!(k " (y "1))!

=k!(y "1)!(k " (y "1))!+k!y!(k " y)!y!(k " y)!(y "1)!(k " (y "1))!

=((y "1)!(k " (y "1))!+y!(k " y)!)k!y!(k " y)!(y "1)!(k " (y "1))!

=((k " y +1)!+y(k " y)!)k!y!(k " y)!(k " y +1)!

=((k " y +1) + y)k!y!(k " y +1)!

=(k +1)k!

y!((k +1) " y)!=

(k +1)!y!((k +1) " y)!

=k+1Cy

!

LHS = ak+1b0+(kC1+kC0)akb1+(kC2+kC1)a

k"1b2+(kC3+kC2)ak"2b3 ...

+(kCk"1+kCk"2)a2bk"1+(kCk+kCk"1)a

1bk+a0bk+1

= ak+1b0+k+1C1akb1+k+1C2a

k"1b2+k+1C3ak"2b3 ...

k+1Ck"1a2bk"1+k+1Cka

1bk+a0bk+1

= (k+1Cr)r= 0

k+1

# a(k+1)"rbr

LHS = RHS

Since S(k) implies S(k+1), by the principles of Mathematic Induction, S(n) is true for all positive integers.

2.2. Examples with expansion of different multinomials and conjecture

Although the trinomial theorem is the expansion of multinomial with three variables, it can also be expressed as a binomial, by treating two of the variables as one (Dhand). For example:

!

(a1 + a2 + a3)n = ((a1 + a3) + a2)

n As a result, the equation can be expanded treating two of the variables as 1:

!

((a1 + a3) + a2)n = nCr1

(a1 + a3)n"r1 a2

r1

r1 =0

n

# = nCr1( n"r1

Cr2r2 =0

n"r1

# a1n"r1 "r2a3

r2 )a2r1

r1 =0

n

#

= (nCr1)(n"r1Cr2

)a1n"r1 "r2a3

r2a2r1

r2 =0

n"r1

#r1 =0

n

#

In other words, the expansion of a trinomial can be expressed as a “binomial within a binomial”. The order of the summation and the combination can be swapped since that particular combination

!

(nCr1) is not affected by the second summation involving

!

r2 . As a result, the combination can be treated more or less like a constant relative to the second summation; because of that, it does not make a difference whether the combination is before or after that summation (Stat 23400: Mathematics Tutorial). The coefficient of the variable can be further simplified:

!

=(nCr1)(n"r1Cr2

) =n!

(n " r1)!r1!(n " r1)!

(n " r1 " r2)!r2!=

n!r1!(n " r1 " r2)!r2!

When multiplying the two combinations together,

!

(n " r1)! from the numerator of the second combination and

!

(n " r1)! from the denominator of the first numerator cancel out. As a result, the factorial of the variables in the denominator are seen to be the same as the exponents of the variables a, b and c. As a result, the coefficient of the variables can be rewritten as:


4

!

=r2 =0

n"r1

#r1 =0

n

# (nCr1)(n"r1Cr2

)a1n"r1 "r2a3

r2a2r1 =

r2 =0

n"r1

#r1 =0

n

# n!r1!(n " r1 " r2)!r2!

a1n"r1 "r2a3

r2a2r1

Let n1, n2, n3...nm be the exponent of a1, a2, a3 ...am respectively. Thus (n " r1 " r2) = n1, r1 = n2, r2 = n3.

=r2 =0

n"r1

#r1 =0

n

# n!n2!n1!n3!

a1n1a3

n3a2n2 =

r2 =0

n"r1

#r1 =0

n

# n!n1!n2!n3!

a1n1a2

n2a3n3

This simplifies the rather complicated “binomial within a binomial” formula into one that is much more manageable. The same process can be used for a multinomial with 4 terms: The expansion of a multinomial with 4 terms has two terms within both of the two terms:

!

((a1+a2+a3+a4 )n = ((a1+a3) + (a2+a4 ))

n = nCr1(a1+a3)

n"r1 (a2+a4 )r1

r1=0

n

#

= nCr1( n"r1

Cr2r 2

n"r1

# a1n"r1"r2a3

r2 )( r1Cr3

r 3

r1

# a2r1"r3a4

r3 )r1=0

n

#

= (nCr1)(n"r1Cr2

)(r1Cr3)a1

n"r1a3r2

r3 =0

r1

#r2 =0

n"r1

#r1 =0

n

# a2r1"r3a4

r3

the coefficient can once again be simplified:

!

=(nCr1)(n"r1Cr2

)(r1Cr3) =

n!(n " r1)!r1!

(n " r1)!(n " r1 " r2)!r2!

r1!(r1 " r3)!r3!

=n!

(n " r1 " r2)!r2!(r1 " r3)!r3!Using the same technique as the one used for the expansion of the trinomial, the

!

(n " r1)! in the numerator and denominator cancel out, as well as the

!

r1! in the numerator and denominator. As a result, the variables in the denominator can be seen as the same as the exponents as the variables a1, a2, a3 and a4. the coefficient of the variables can be rewritten as:

!

((a1 + a3) + (a2 + a4 ))n =

n!(n " r1 " r2)!r2!(r1 " r3)!r3!

a1n"r1"r2a3

r2

r3 =0

r1

#r2 =0

n"r1

#r1 =0

n

# a2r1"r3a4

r3

=n!

n1!n3!n2!n4!a1n1a3

n3a2n 2a4

n4

r3 =0

r1

#r2 =0

n"r1

#r1 =0

n

#

=n!

n1!n2!n3!n4!a1n1a2

n 2a3n3a4

n4

r3 =0

r1

#r2 =0

n"r1

#r1 =0

n

#

Based on the patterns from the expansion of the trinomial and quadnomial, I predict that:


5

!

(a1 + a2 + a3 + ...am )n =

n!n1!n2!n3!n4!...nm!

a1n1a2

n2a3n 3 ...am

nm

n1 ,n2 ,n3 ...nm

"

For all possible integers of

!

n1,n2,n3,n4 ...nm , such that

!

n1 +n2 + n3 + n4 + ...nm = n (Dhand), where

!

n1 ,n2 ,n3 ...nm

" means that for every possible combination of

!

n1,n2,n3...nm , there will be

another term to add.

2.3. Proof of Multinomial Theorem with Mathematical Induction

The Multinomial Theorem can be proved with a combination of the Mathematical Induction and the Binomial Theorem (Dhand): Let S(m) be the statement:

!

(a1 + a2 + a3 + ...am )n =

n!n1!n2!n3!n4!...nm!

a1n1a2

n2a3n 3 ...am

nm

n1 ,n2 ,n3 ...nm

"

For all possible values of

!

n1,n2,n3,n4 ...nm , such that

!

n1 +n2 + n3 + n4 + ...nm = n For

!

m =1:

!

LHS = (a1)n = a1

n

RHS =n!n1!

!

n1 = n , as there is only one variable, and as a result only one exponent that makes up n.

!

RHS = a1n

LHS = RHS

S(1) is true For

!

m = 2

!

LHS = (a1 + a2)n

Through the use of the binomial theorem:

!

LHS = (a1 + a2)n = (nCr1

)r1 =0

n

" a1n#r1a2

r1 =n!

n # r1!r1!r1 =0

n

" a1n#r1a2

r1

The denominator of the coefficient is equivalent to the factorial of the exponents of the variables a1 and a2. As a result,

!

(n " r1) = n1 and

!

r1 = n2 .

!

LHS =n!

n1!n2!r1 =0

n

" a1n1a2

n2

!

RHS =n!

n1!n2!a1n1a2

n2 for all possible values of

!

n1,n2 , such that

!

n1 +n2 = n

!

LHS = RHS S(2) is true Let’s assume S(m) is true for some value m, say k, that is


6

!

(a1 + a2 + a3 + ...ak )n =

n!n1!n2!n3!n4!...nk!

a1n1a2

n2a3n 3 ...ak

nk

n1 ,n2 ,n3 ...nk

"

for all possible values of

!

n1,n2,n3,n4 ...nk , such that

!

n1 +n2 + n3 + n4 + ...nk = n For

!

m = k +1:

!

LHS = (a1 + a2 + a3 + ...ak + ak+1)n

let’s treat “

!

ak + ak+1” as 1 variable, say

!

aw , where

!

aw is still the kth term, although the value of

!

aw is different from

!

ak.

!

LHS = (a1 + a2 + a3 + ...aw )n

Since the number of variables is still k, the equation for k can be used.

!

LHS =n!

n1!n2!n3!n4!...nw!a1n1a2

n2a3n 3 ...aw

nw

n1 ,n2 ,n3 ...nw

" For all possible values of

!

n1,n2,n3,n4 ...nw , such that

!

n1 +n2 + n3 + n4 + ...nw = n

!

LHS =n!

n1!n2!n3!n4!...nw!a1n1a2

n2a3n 3 ...(ak + ak+1)

nw

n1 ,n2 ,n3 ...nw

"

!

LHS =n!

n1!n2!n3!n4!...nw!a1n1a2

n2a3n 3 ...( nw

Cnk+1ak

nw "nk+1ak+1nk+1

nk+1

nw

# )n1 ,n2 ,n3 ...nw

# for all

possible values of

!

n1,n2,n3,n4 ...nw " nk+1,nk+1, such that

!

n1 +n2 + n3 + n4 + ...nw " nk+1 + nk+1 = n

!

LHS =n!

n1!n2!n3!n4!...nw!nwCnk+1

a1n1a2

n2a3n 3 ...

nk+1

nw

" aknw #nk+1ak+1

nk+1

n1 ,n2 ,n3 ...nw

"

=n!

n1!n2!n3!n4!...nw!nw!

nw # nk+1!nk+1!a1n1a2

n2a3n 3 ...

nk+1

nw

" aknw #nk+1ak+1

nk+1

n1 ,n2 ,n3 ...nw

"

=n!

n1!n2!n3!n4!...nw # nk+1!nk+1!a1n1a2

n2a3n 3 ...

nk+1

nw

" aknw #nk+1ak+1

nk+1

n1 ,n2 ,n3 ...nw

"

!

nw " nk+1 = nk

!

LHS =n!

n1!n2!n3!n4!...nk!nk+1!a1n1a2

n2a3n 3 ...ak

nk ak+1nk+1

n1 ,n2 ,n3 ...nk ,nk+1

" For all possible values

of

!

n1,n2,n3,n4 ...nk,nk+1, such that

!

n1 +n2 + n3 + n4 + ...nk + nk+1 = n

!

LHS = RHS S(k+1) is true Since S(k+1) implies S(k), according to the principles of mathematical induction, S(m) is true.

2.3.1. Mechanism for Multinomial Theorem

Although Mathematical Induction effectively proves the Multinomial Theorem, it does not provide a mechanism for why it works.


7

In a binomial coefficient, there is a set of elements whose factorial is the numerator. The denominator is the factorial of a certain amount of elements from the original set, multiplied by the factorial of the difference between the original set and the new amount of elements. A multinomial can be expressed as “layers of binomials”; the exponent of the “outer layer”, say layer a, is the original set of elements, whilst there are two new amount of elements formed in an “inner layer”, say layer b; one of these number of elements can be seen as the “new amount of elements”, whilst the other can be seen as the “difference between the original set and the new amount of elements”. As mentioned previously, the factorials of the new amount of elements, in the case of a binomial, the exponents, are in the denominator. This explains the mechanism for the binomial theorem. For the multinomial theorem, the “inner layers” are divided further and further. However, the new amount of elements, or the exponent for a binomial in layer b becomes the total number of elements in regards to that binomial. Two new number of elements are once again formed, in another inner layer, say layer c, one which is the “new amount of elements from layer c”, whilst the other is the “difference between the original set of elements from layer b and the new amount of elements from layer c”. Since one of the new smaller set of elements from layer b is the original set of elements for layer c, it appears in both the numerator and the denominator and cancels itself out. As a result, it only leaves the factorial of the total set of elements from the first layer (the original exponent) divided by the factorials of the new set of elements (the exponents of each variable).

3. Multinomial Theorem with combinatorics

Just as the Binomial Coefficient represents the number of unordered ways to pick a k-subset of elements from a set of n elements, the Multinomial Coefficient can be used to express probabilities and different number of outcomes as well (Combination -- from Wolfram MathWorld). One commonly used example revolves around the number of outcomes for rearranging words (Dhand); for example, in the word “SENSELESSNESS”. The total number of letters in the word can be treated as the set, whilst each letter can be viewed as a subset (The multinomial theorem | A Blog on Probability and Statistics). There is a total of thirteen letters in the word. However, because of the number of repeated words, the amount of possible outcomes is much less than

!

13!. There are 6 S’s, 4 E’s, 2 N’s and 1L, and using the multinomial coefficient, the number of possible ways to rearrange the word is:

!

13!6!4!2!1!

=180180 different ways.

!

a1 + a2

!

a1 + a3

Layer a

Layer b

Figure 1 Diagram of different layers


8

4. Multinomial Theorem and Combinatorics through geometry

4.1. Pascal’s Triangle

The binomial coefficients create a unique pattern that can be represented through Pascal’s triangle, seen in figure 2 (KryssTal : Pascal's Triangle). The left and right side consists of the number 1, whilst the rest of the numbers are the sum of the two numbers directly above it. The numbers also happen to be the binomial coefficients, where the nth row of numbers represents the exponent n from the binomial. This pattern can be explained with Pascal’s Rule (PlanetMath: Pascal's rule). This can be seen in the proof below:

!

nCr+nCr+1 =n!

(n " r)!r!n!

(n " (r +1))!(r +1)!=n!((n " (r +1))!(r +1)!+(n " r)!r!)((n " (r +1))!(r +1)!(n " r)!r!)

=n!((n " r "1)!(r +1)!+(n " r)!r!)((n " r "1)!(r +1)!(n " r)!r!)

=n!((n " r "1)!(r +1)r!+(n " r)(n " r "1)!r!)

((n " r "1)!(r +1)!(n " r)!r!)

=n!((r +1) + (n " r))((n " r "1)!r!)((n " r "1)!(r +1)!(n " r)!r!)

=n!(n +1)

(r +1)!(n " r)!=

(n +1)!(r +1)!((n +1) " (r +1))!

=n+1Cr+1

It is comparable to a portion of the mathematical induction proof of the binomial theorem, and it demonstrates the relationship between Pascal’s triangle and the binomial theorem.

4.2. Pascal’s Pyramid

The pattern of binomial coefficients in Pascal’s triangle can also be extended to trinomial coefficients (Horn 1-2). The coefficients of a trinomial can be demonstrated in a form that resembles a pyramid, which is known as Pascal’s Pyramid. The pyramid can be divided into levels, where the nth level represents the coefficients of a trinomial with an exponent n, as seen in figure 3 (Bondarenko 48). Whereas in the Pascal’s triangle the left and right side were all ones, in the Pascal’s Pyramid, the vertices of the pyramid are 1. For each level, which is shaped like a triangle, the numbers on the side are the binomial coefficients, and as a result there are three sets of binomial coefficients for the three sides, other than the ones, for which there are only three rather than six. For the rest of the trinomial coefficients, they appear in the middle of each level, and are the sum of the three numbers right above it, as seen in figure 4 (Horn 1-2).

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Figure 2 Pascal’s Triangle


9

1

1 1

1

1 1

2 2

2 1 1

1

3

3

3 3

3

3 6

1

1 1

4

4

4

4

4

4

6

6

6 12

12 12

1

1 1

5

5

5 5

5

5

10

10

10

10

20

20

20

30

30

30

2+2+2=6

3+3+6=12 12+4+4=12

6+12+12=30

Figure 3 “Levels” of the Pascal Pyramid (Bondarenko 48). The sides, the binomial expansion equivalent, are black, whilst the “new” coefficients are red.


10

1

1 1 1

1

1 1 1

1 1

1

1

2 2

2 3

3

3 3

3 3 6

4 6

4

4 6 4

4

4

6 12 12 12

Figure 4 Pascal’s Pyramid (Horn 1-2). The odd levels are red whilst the even levels are black.


11

4.3. Pascal’s Hyper-Pyramids

The patterns of multinomial coefficients where a multinomial has greater terms than a binomial and trinomial can also be expressed geometrically (Farkas and Kallos 109-110). However, as the binomial has a corresponding 2-dimensional shape, and the trinomial has a corresponding 3-dimensional shape, the pattern shows that the quadnomial would most likely correspond to a 4-dimensional shape, a 5-term multinomial to a 5-dimensional shape, and a mth-term multinomial to a m-dimensional shape (Pascal's Simplices). However, due to natural limitations in visualization, it is extremely difficult to imagine a shape more than the natural three dimensions. The table below shows what the pattern so far indicates the shapes will most likely look like:

Number of terms in multinomial

1 2 3 4 5 … m

Number of Dimensions

1 2 3 4 5 … m

Shape Line Triangle Tetrahedron 5-tetrahedron

6(5-tetrahedron)

…

!

(m +1) Previous shape

2 points 3 sides (lines)

4 triangles 5 tetrahedrons

6 (5-tetrahedrons)

…

!

(m +1) Previous shape

The table suggests that the corresponding shape to the number of terms in a multinomial, m, is

!

(m +1) sets of the previous shape. However, that still does not show exactly what the shape would look like, as they would be put together in a way that cannot be visualized by the human mind, due to the additional dimensions. For example, even though the tetrahedron is made up of 4 triangles, it is a very specific construction of the 4 triangles, in a 3-dimensional space rather than just putting the 4 triangles side by side. Just as a being operating in 2-dimensions would not be able to see the tetrahedron, I am unable to see the shapes for terms in a multinomial greater than 3.

4.4. Summation of row and layers with Pascal’s Hyper-Pyramids

There are also a multitude of trends and patterns with Pascal’s Hyper-Pyramids (Pascal's Triangle And Its Patterns). One of which is that for the Pascal’s triangle, the sum of all the numbers in the nth row results in 2n (Pascal’s Simplices Project).

Figure 5 Table of number of terms in multinomial and expected shape of hyper-pyramid


12

This can be proved through the use of mathematical induction:

Let S(n) be the statement:

!

nC0+nC1+nC2 ...n Cn = 2n for all

!

n"Z + for

!

n =1:

!

LHS=1C0+1C1 =1+1 = 2RHS = 2(1) = 2LHS = RHS

S(1) is true.

Lets assume S(n) is true for some value k, that is:

!

kC0+kC1+kC2 ...k Ck = 2k for all

!

k "Z + for

!

n = k +1:

!

LHS=k+1C0+k+1C1+k+1C2 ...k Ck+k+1Ck+1

=k+1C0 +(kC0 +k C1) +(kC1 +k C2) +(kC2 +k C3)...+(kCk"1 +k Ck )+k+1Ck+1

As proved previously, the combination

!

x+1Cy+1 =x Cy +x Cy+1. Using this proof, it can be seen that there are two of every

!

kCn , as long as n is greater than 0 and less than k. This is because

!

k+1Cn will produce a

!

kCn , whilst

!

k+1Cn+1will also produce a

!

kCn . However, this is not true for

!

kC0 and

!

k Ck since

!

k+1C0 and

!

k+1Ck+1 cannot split to 2, as it would result in a combination where a negative amount of elements is being picked, and a combination where the amount of elements being picked exceeds the amount of elements in the set. As a result the equation can be reformatted:

!

LHS=k+1C0+kC0 + 2(kC1 +k C2 +k C3 ...+kCk"1) +k Ck+k+1Ck+1 However, since

!

k+1C0=

!

kC0 =

!

kCk=

!

k+1Ck+1=1, the

!

k+1C0 and

!

k+1Ck+1 can be substituted with

!

kC0 and

!

kCk respectively:

Figure 6 Summation of nth row in Pascal’s Triangle

!

1 =1 = 20

1+1 = 2 = 21

1+ 2 +1 = 4 = 22

1+ 3+ 3+1 = 8 = 23

1+ 4 + 6 + 4 +1 =16 = 24

1+ 5 +10 +10 + 5 +1 = 32 = 25

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1


13

!

LHS = 2(kC0 +k C1 +k C2 +k C3 ...+kCk"1 +k Ck ) = 2(2k ) = 2k+1 = RHS Because LHS=RHS, S(k) is true.

Since S(k) implies S(k+1), by the principles of Mathematic Induction, S(n) is true for all positive integers.

This statement can also be extended to the other degrees of Pascal’s Hyper-Pyramids as well. For a multinomial

!

(a1 +a2+a3 + ...+ am )n , the sum of the coefficients are

!

mn (Pascal's Simplices), as seen in a simple explanation I discovered online after my original proof for the sum of coefficients in Pascal’s Triangle:

Since in the factorized form, the coefficients of each of the variables are 1, the sum of the coefficients can be interpreted as

!

(1+1+1+ ...+1)n . Since there are m variables, the total sum of the coefficients is

!

(m)n . 5. Multinomial Theorem & Combinatorics with real life application (poker

dice)

One possible application of the multinomial coefficients is through the game of poker dice (The game of poker dice and the multinomial theorem). Poker dice is similar to the game of poker but is played with 5 dice instead of cards. The 5 dice each have 6 possible numbers, and the combination it forms are ranked from lowest to highest, where the player with the highest score is the winner:

Single- all dice have different numbers Pair- 1 number appears on two dice 2 Pairs- 2 numbers appear twice Triple- 1 number appears on three dice Full House- 1 Triple and 1 Pair 4 of a kind- 1 number appears on four dice 5 of a kind- 1 number appears on 5 dice

There are 65 possible results, including orders, since each die has 6 different possibilities. Using the multinomial theorem, it is possible to calculate the probability of each combination. For a triple, it would require 3 of 1 number and 2 other different numbers, regardless of the number. One example of this can be seen below:

1 2 3 4 5 6

3 1 1 0 0 0

As a result, it would require some order of these numbers. In order to avoid confusion, let’s label the boxes A, B, C, D, E and F in order to avoid confusion:

A B C D E F

3 1 1 0 0 0

Regardless of what numbers the letters are, a triple would require some order of the numbers 3, 1, 1, 0, 0, 0. By using the multinomial coefficient twice in a row, I can


14

calculate the possibility of 1 particular triple with two particular singles, say a triple of A with singles B and C, and extend that to include all other pairs. For the calculation of the possible outcomes with triples of A with singles B and C, the use of the multinomial coefficient similar to the use in part 3 is used. Since there is 1 number repeated 3 times, the possible outcomes of 3 A’s, 1 B and C will be:

!

5!3!1!1!

= 20

In order to calculate the other possible outcomes of 1 triple and 2 singles, the multinomial coefficient needs to be applied again, but with regards to the amount of singles, amount of pairs, amount of triples… etc. Since there is 1 triple, 2 singles and 3 numbers that do not appear, the equation applied is:

!

6!1!2!3!

= 60

Since there are 60 possible outcomes of the 20 possible outcomes, the total possible outcomes for 1 triple is:

!

60 " 20 =1200 As a result, the possibility of a triple is:

!

(12007776

) "100% =15.432%

Using this method, the amount of the other combinations can be calculated as well: Single:

A B C D E F

1 1 1 1 1 0

!

5!1!1!1!1!1!"

# $

%

& ' 6!5!1!"

# $

%

& ' (100% = 9.259%

1 Pair:

A B C D E F

2 1 1 1 0 0

!

5!2!1!1!1!"

# $

%

& '

6!1!3!2!"

# $

%

& ' (100% = 46.296%

2 Pairs:

A B C D E F

2 2 1 0 0 0

!

5!2!2!1!"

# $

%

& '

6!2!1!3!"

# $

%

& ' (100% = 23.148%

Full House:

A B C D E F

3 2 0 0 0 0


15

!

5!3!2!"

# $

%

& ' 6!1!1!4!"

# $

%

& ' (100% = 3.858%

4 of a Kind:

A B C D E F

4 1 0 0 0 0

!

5!4!1!"

# $

%

& ' 6!1!1!4!"

# $

%

& ' (100% =1.929%

5 of a Kind:

A B C D E F

5 0 0 0 0 0

!

5!5!"

# $

%

& ' 6!1!5!"

# $

%

& ' (100% = 0.077%

By adding up the total amount of percentages, it can also be seen that every possible combination is taken into consideration:

!

15.432% + 9.259% + 46.296% + 23.148% + 3.858% +1.929% + 0.077% = 99.999% (Accurate to 3 decimal points).

6. Conclusion

The Multinomial Theorem is an extension of the binomial theorem, and as a result has many implications in both pure and applied mathematics. In my extended essay, I answer the question “What is the multinomial theorem, what is its relationship with combinatorics and how can it be applied to both real and abstract fields?” Through proof by mathematical induction, I proved that the multinomial theorem is:

!

(a1 + a2 + a3 + ...am )n =

n!n1!n2!n3!n4!...nm!

a1n1a2

n2a3n 3 ...am

nm

n1 ,n2 ,n3 ...nm

"

I discussed applications of the multinomial theorem and combinatorics to abstract fields, such as Pascal’s Hyper-Pyramids, whilst also investigated the real applications of the multinomial theorem and combinatorics in the form of Poker dice. With more developed knowledge and a higher understanding of mathematics, the multinomial theorem could answer unresolved questions regarding Pascal’s Hyper-Pyramids, explaining the shapes more clearly and the layers within these shapes. It can also be extended to areas that are outside the realms of traditional pure mathematics and lead to new questions: What are the applications of the multinomial theorem to game theory, cryptography and computer science? These are all areas that could be inter-related to the multinomial theorem. As a result, it is a topic that I intend to pursue as I continue to learn mathematics, in high school and at the university I attend.


16

7. Works Cited

"Binomial theorem - Topics in precalculus." TheMathPage: math lessons and math

help. N.p., n.d. Web. 16 June 2011.

<http://www.themathpage.com/aprecalc/binomial-theorem.htm#theorem>.

Bona, Miklos. A walk through combinatorics: an introduction to enumeration and

graph theory. 2nd ed. Singapore: World Scientific, 2006. Print.

Bondarenko, Boris. Generalized Pascal triangles and pyramids: their fractals,

graphs, and applications. Aurora, SD: Fibonacci Assoc., 1993. Print.

"Combination -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most

Extensive Mathematics Resource. N.p., n.d. Web. 8 June 2011.

<http://mathworld.wolfram.com/Combination.html>.

"Combinatorics -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most

Extensive Mathematics Resource. N.p., n.d. Web. 14 June 2011.

<http://mathworld.wolfram.com/Combinatorics.html>.

Dhand, Vivek. "NOTES FOR MATH 481 MULTINOMIAL COEFFICIENTS."

Michigan State University Mathematics. N.p., n.d. Web. 22 June 2011.

<www.math.msu.edu/~dhand/MTH481FS08/Notes7.pdf>.

Farkas, Gabor, and Gabor Kallos. "Prime Number in Generalized Pascal Triangles."

Acta Technica Jaurinensis 1.Nov 1st (2008): 109-118. Acta Technica

Jaurinensis. Web. 14 June 2011.

Horn, Martin Erik . "Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral

Multinomial Theorem." arxiv.org 1 (2003): 1-2. arXiv.org. Web. 7 June

2011.

"KryssTal : Pascal's Triangle." KryssTal : Home Page. N.p., n.d. Web. 14 June 2011.

<http://www.krysstal.com/binomial.html>.


17

"Multinomial Series -- from Wolfram MathWorld." Wolfram MathWorld: The Web's

Most Extensive Mathematics Resource. N.p., n.d. Web. 13 June 2011.

<http://mathworld.wolfram.com/MultinomialSeries.html>.

"Pascal's Simplices." Mathematics Department - Welcome. N.p., n.d. Web. 20 July

2011. <http://www.math.rutgers.edu/~erowland/pascalssimplices.html>.

"Pascal's Simplices Project." Mathematics Department - Welcome. N.p., n.d. Web. 17

July 2011. <http://www.math.rutgers.edu/~erowland/pascalssimplices-

project.html>.

"Pascal's Triangle And Its Patterns." All You Ever Wanted to Know About Pascal's

Triangle and more. N.p., n.d. Web. 19 July 2011. <http://ptri1.tripod.com/>.

"PlanetMath: Pascal's rule." PlanetMath. N.p., n.d. Web. 15 June 2011.

<http://planetmath.org/encyclopedia/PascalsRule.html>.

"Stat 23400: Mathematics Tutorial." The University Of Chicago- The Department of

Statistics. N.p., n.d. Web. 3 Oct. 2011.

<galton.uchicago.edu/~stat234/prereq/math_tutorial.pdf>.

"The game of poker dice and the multinomial theorem." A Blog on Probability and

Statistics. N.p., n.d. Web. 16 June 2011.

<http://probabilityandstats.wordpress.com/2010/04/28/the-game-of-poker-

dice-and-the-multinomial-theorem/>.

"The multinomial theorem | A Blog on Probability and Statistics." A Blog on

Probability and Statistics. N.p., n.d. Web. 16 June 2011.

<http://probabilityandstats.wordpress.com/2010/04/24/the-multinomial-

theorem/>.

"Wolfram|Alpha: Math Reference: binomial." Wolfram|Alpha: Computational

Knowledge Engine. N.p., n.d. Web. 15 June 2011.


18

<http://www.wolframalpha.com/entities/mathworld/binomial/op/i4/ps/>.

Documents

Extended Essay- Mathematics - GitHub Pagesoscarw3.github.io/files/wang_oscar_EE_mathematics.pdf · Oscar Wang Chinese International School Candidate No.: 0637 095 Extended Essay-