10.2 Combinations and Binomial Theorem
What you should learn:
GoalGoal 11
GoalGoal 22
Use Combinations to count the number of ways an event can happen.
Use the Binomial Theorem to expand a binomial that is raised to a power.
10.2 Combinatins and Binomial Theorem10.2 Combinatins and Binomial Theorem
In the last section we learned counting problems where
order was important• For other counting problems where
order is NOT important like cards, (the order you’re dealt is not important, after you get them, reordering them doesn’t change your hand)
• These unordered groupings are called Combinations
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
A Combination is a selection of r objects from a group of n objects
where order is not important
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Combination of n objects taken r at a time
• The number of combinations of r objects taken from a group of n distinct objects is
denoted by nCr and is:
!)!(
!
rrn
nCrn
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
For instance, the number of combinations of 2 objects taken from a group of 5 objects is
101*2*1*2*3
1*2*3*4*5
!2)!25(
!525
C
2
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Finding Combinations
• In a standard deck of 52 cards there are 4 suits with 13 of each suit.
• If the order isn’t important how many different 5-card hands are possible?
• The number of ways to draw 5 cards from 52 is
!5!*47
!47*48*49*50*51*52
!5)!552(
!52552
C
= 2,598,960
In how many of these hands are all 5 cards the same suit?
• You need to choose 1 of the 4 suits and then 5 of the 13 cards in the suit.
• The number of possible hands are:
5148!5!*8
!8*9*10*11*12*13*
!1!*3
!3*4
!5!*8
!13*
!1!*3
!4* 51314 CC
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
How many 7 card hands are possible?
• How many of these hands have all 7 cards the same suit?
560,784,133!7!*45
!52752 C
6864* 71314 CC
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
When finding the number of ways both an event A and an event B can occur, you multiply.
When finding the number of ways that an event A OR B can occur, you +.
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Deciding to ADD or MULTIPLY
A restaurant serves omelets. They offer 6 vegetarian ingredients and 4 meat ingredients.
You want exactly 2 veg. ingredients and 1 meat. How many kinds of omelets can you order?
604*15!1!3
!4*
!2!4
!6* 1426 CC
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Suppose you can afford at most 3 ingredients
How many different types can you order?
You can order an omelet with 0, or 1, or 2, or 3 items and there are 10 items to choose from.
17612045101310210110010 CCCC
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Counting problems that involve ‘at least’ or ‘at most’ sometimes
are easier to solve by subtracting possibilities you
don’t want from the total number of possibilities.
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Subtracting instead of adding:
A theatre is having 12 plays. You want to attend at least 3. How many combinations of plays can you attend?
•You want to attend 3 or 4 or 5 or … or 12.
•From this section you would solve the problem using:
•Or……1212512412312 ... CCCC
For each play you can attend you can go or not go.
•So, like section 10.1 it would be 2*2*2*2*2*2*2*2*2*2*2*2 =212
•And you will not attend 0, or 1, or 2.
•So:4017)66121(4096)(2 212112012
12 CCC
0C0
1C0 1C1
2C0 2C1 2C2
3C0 3C1 3C2 3C3
4C0 4C1 4C2 4C3 4C4
Etc…
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Pascal's Triangle!
• 1• 1 1
• 1 2 1• 1 3 3 1
• 1 4 6 4 1• 1 5 10 10 5 1
• Etc…• This describes the coefficients in the
expansion of the binomial (a+b)n
• (a+b)2 = a2 + 2ab + b2 (1 2 1)
• (a+b)3 = a3(b0)+3a2b1+3a1b2+b3(a0) (1 3 3 1)
• (a+b)4 = a4+4a3b+6a2b2+4ab3+b4 (1 4 6 4 1)
• In general…
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
(a+b)n (n is a positive integer)=
• nC0anb0 + nC1an-1b1 + nC2an-2b2 + …+ nCna0bn
• =
n
r
rrnrn baC
0
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
(a+3)5 =
• 5C0a530+5C1a431+5C2a332+5C3a233+
5C4a134+5C5a035=
• 1a5 + 15a4 + 90a3 + 270a2 + 405a + 243
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem
Assignment
12.2 Combinatins and Binomial Theorem12.2 Combinatins and Binomial Theorem