Factorials and Their Applications

I.  Homework

II.  Definition of the Factorial

We define n! recursively by

0! = 0, 1! = 1, n! = n(n - 1)!

Example:  5! = 5(4)(3)(2) = 120

III.  Permutations

Suppose that we are interested in how many ways there are in scrambling the letters of the name "Cindy".  We have 5 choices for the first letter, once we have chosen the first letter there are 4 choices for the second letter, and then three choices for the third letter, two for the fourth letter, and only one choice for the last letter.  Hence there are

5(4)(3)(2)(1) = 5! choices.

If we want to select only three letters then we have (5)(4)(3) = 5!/(5 - 3)! choices.

Definition:  The number of permutations of n distinct objects taken r at a time is

nPr = n!/(n - r)!

You can find this button on the TI 85 calculator by hitting Math -> Prob

IV  Distinguishable Permutations

How many ways are there of scrambling the name Tamara Heether?

Solution:  If there were no duplicate letters the solution would be 13!, but this is not the case.  There are 2 T's, 3 A's 2 R's and 3E's.

We must divide by 2!3!2!3! to get

13!/2!3!2!3! = 43,243,200 

Theorem:  If there are n objects with n₁ duplicates of one kind, n₂ duplicates of a second kind, ..., n_k duplicates of a kth kind, then  the number of distinguishable permutations of these n objects is

n!/(n₁!n₂!...n_k!)

Exercise:  How many ways are there to scramble your first and last name?

V.  Combinations

Example:  How many different five card poker hands are there?

First note that there are 52P5 different ordered five card poker hands, but two hands that have the same five cards, but in different order should not be counted as distinct hands.  Since there are 5! ways of ordering five cards, we have

52P5/5! = 52!/[5!(52 - 5)!] = 2,598,960

different poker hands.

Note that only four of these hands are Royal Flushes, hence there is a 4 in 2,598,960 or about one in half a million chance of receiving a Royal Flush in a 5 card stud poker game.

Theorem:  The number of ways of choosing r objects from n where order does not matter is

nCr = n!/(n - r)!r!

VI.  The Binomial Theorem

consider (x + y)⁵  = (x + y)(x + y)(x + y)(x + y)(x + y)

Q:  How many ways are there to select all x's?

A:  1 way

Q:  How many ways are there to select 4x's from the 5?

A:  5C4

Exercise: How many ways are there to select two x's from the five?

Hence

(x + y)⁵  = 5C5 x⁵  + 5C4 x⁴y + 5C3 x³y² + 5C2 x²y³ + 5C1 xy⁴ 5C0 y⁵  

Theorem:  

(x + y)ⁿ = sum from i = 0 to n of nC(n - i) x^n-iyⁱ

Exercise:  Find (3x - 2y)⁴