Permutation & Combination

Introduction

By considering the ratio of the number of desired subsets to the number of all possible subsets for many games of chance in the 17th century, the French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory.

Important Terms: Factorial

 The continued product of first n natural numbers is called the “n factorial”.  It is denoted by an exclamatory symbol like n! or |n.

Examples: n! = n × (n – 1) × … × 4 × 3 × 2 × 1  5! = 5 × 4 × 3 × 2 × 1 = 120

Factorial is defined for whole numbers only.Factorials of proper fractions or negative integers are not defined.Factorial of zero is defined to be 1. That is, 0! = 1. This is consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects, that is, there is a single permutation of zero elements, namely the empty set φ.

Fundamental Principle of Counting:  (also known as the multiplication rule for counting)  If a task can be performed in n1 ways, and for each of these a second task can be performed in n2 ways, and for each of the latter a third task can be performed in n3 ways, ..., and for each of the latter a kth task can be performed in nk ways, then the entire sequence of k tasks can be performed in  n1 • n2 • n3 • ... • nk  ways.

Permutation

The origin of permutation is the Latin word mutare which related to change.  The per prefix is an emphsis of complete or extreme change. 

A set of objects in which position (or order) is important. A permutation which only exchanges two elements is often

called a transposition.  It is known that every permutation can be found from any other by a string of transpositions.  The notation to switch the first and third element in a set is (1,3). 

A permutation is the choice of r things from a set of n things without replacement and where the order matters.

Two types of Permutation:

Repetition is Allowed: such as the lock above. It could be "333".

No Repetition: for example the first three people in a running race. You can't be first and second.

1. Permutations with Repetition

These are the easiest to calculate.When we have n things to choose from ... we have n choices each time!When choosing r of them, the permutations are:

n × n × ... (r times)

So, the formula is simply:nr,

where n is the number of things to choose from, and we choose r of them(Repetition allowed, order matters)

2. Permutations without Repetition

where n is the number of things to choose from, and we choose r of them(No repetition, order matters)

Permutations in a circle:

Sometimes, items are arranged in a circle. In this case, we no longer have a left end and a right end. The first placed item is merely a point of reference instead of having n choices.

Thus with n distinguishable objects we have (n-1)! arrangements instead of n!.

PERMUTATION EXAMPLES:1. How many ways can we award a 1st, 2nd and 3rd place prize among eight contestants? (Gold / Silver / Bronze)?

Let’s say:Gold medal: 8 choices: A B C D E F G HSilver medal: 7 choices: B C D E F G H. Bronze medal: 6 choices: C D E F G H.

Using this formula:

So the total number of options is 336

2. Permutation without Repetitions: What is the total number of possible 4-letter arrangements of the letters m, a, t, h if each letter is used only once in each arrangement?  

Using the formula: Using calculator:

3. How many 3-letter word with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?The word 'LOGARITHMS' has 10 different letters.

Using calculator: 10P3 or P(10,3) = 720 three-letter word can be formed4. Permutation in a Circle: How many six colored squares can be formed in a circle?

C = (n-1)! = (6-1)! = 120

Combination

The word combine, from which combination derives, is built of the Latin prefix com, for together, and bini a special form of the Latin root for two, bi. Bini would be similar to our word pair, and the idea of combining was to put together two at a time.

A set of objects in which position (or order) is NOT important.

A combination is the choice of r things from a set of n things without replacement and where order does not matter. 

Two types of Combinations:

Remember the order does not matter now.

Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combination with Repetition

where n is the number of things to choose from, and we choose r of them(Repetition allowed, order doesn't matter)

COMBINATION EXAMPLES:

1. Combination without repetition: Coleen is on a shopping spree. She buys six tops, three shorts and 4 pairs of sandals.  How many different outfits consisting of a top, shorts and sandals can she create from her new purchases?

Using this formula:

6C1* 3C1* 4C1 = (6)(3)(4) = 72 possible outfits

2. There are 12 boys and 14 girls in Mrs. Schultzkie's math class.  Find the number of ways Mrs. Schultzkie can select a team of 3 students from the class to work on a group project.  The team is to consist of 1 girl and 2 boys.     

Order, or position, is not important.  Using the multiplication counting principle,

3. A box contains 4 red, 3 white and 2 blue balls. Three balls are drawn at random. Find out the number of ways of selecting the balls of different colours?

1 red ball can be selected in 4C1 ways1 white ball can be selected in 3C1 ways1 blue ball can be selected in 2C1 ways

Using calculator: 4C1* 3C1* 2C1 or C(4,1) x C(3,1) x C(2,1) = 24 ways

http://www.pballew.net/permute.html http://www.britannica.com/topic/permutation http://www.regentsprep.org/regents/math/algtrig/ats5/lcomb.htm http://www.cubers.com/classroom/lesson1-2.php https://cbpowell.wordpress.com/2009/11/14/permutations-vs-combinatio

ns-how-to-calculate-arrangements/

https://www.mathsisfun.com/combinatorics/combinations-permutations.html