Permutations and Combinations Exercises 5.3: 4, 6, 12, 16, 30, 40 Exercises 5.5: 4, 10

Permutation A permutation is an ordering of objects For example, 3 blocks can be ordered 6 ways There are n! permutations of n elements Easily proved using the Product rule

r-Permutation r-permutations are used to order subsets of distinct elements P(n,r) represents the r-permutations of a set of n distinct elements P(n,r) = n(n-1)(n-2) … (n-r+1) for r  n P(n,r) = n! / (n-r)! Example There are 8 different candy to choose from. How many different “meals” can you make by choosing 3 of them? P(n,r) = P(8,3) = 8*7*6

r-Combination What if it does not matter which order you eat the candy?... A combination is an unordered arrangement of elements in a set Example: a 3-combination from a set of 12 colored blocks

r-Combination An r-combination is a selection of r unordered elements from a set (or simply a subset with r elements) C(n,r) denotes the number of r-combinations of a set of n elements C(n,r)= P(n,r)/r! = n!/[(n-r)!r!] = C(n,n-r)

Combinations with Repetition Example How many ways are there to select any 4 pieces of fruit from a bowl containing at least 4 apples, 4 bananas and 4 pears assuming selection order does not matter? Solution List the 15 possibilities

Combinations with Repetition Theorem: There are C( n + r - 1, r) r-combinations from a set with n unique elements when repetition of elements is allowed. Previous Example: n = 3 different fruits r = 4 items to select C( 3+4-1, 4) = C(6,4) = 6!/[(6-4)!4!] = 15

Combinations with Repetition Example How many ways are there to select 3 bills from a cash box containing $1, $5, $10, $20 and $50 bills? Solution $1 $5 $10 $20 $50 n = 5 r = 3 C( 3+5-1, 3) C( 7, 3) * * * * * * ** * *** 1 2 3 4

Combination with Repetition Example How many ways can 6 balls be distributed into 9 different bins? Solution n = 9 unique bins r = 6 balls C( 9+6-1, r ) = C(14, 6)

Permutations with Repetition Counting permutations with repetition relies on the multiplication rule Example: How many strings of length r can be formed from the English alphabet? Theorem: The number of r-permutations of a set of n objects with repetition is nr abc abc abc … abc = 26*26*26 … 26 = 26r r r

Example: Which formula to use??? How many different strings can be made by reordering the letters of the string SUCCESS? Solution: 3 S’s, 2 C’s, 1 E and 1 U C(7,3) to place to S’s C(4,2) to place the C’s C(2,1) to place the E C(1,1) to place the U C(7,3)C(4,2)C(2,1)C(1,1) = 7! * 4! * 2! * 1! 3!4! 2!2! 1!1! 0!0!

r-Permutations with Repetition Theorem The number of different permutations of n objects, where there are n1 identical objects of type 1, n2 identical objects of type 2 … and nt identical objects of type t, is n! n1! n2! …nt!

Permutations & Combinations For r elements selected from n distinct elements: r-permutations P(n,r) r-combinations C(n,r) n! (n-r)! n! (n-r)!r! r-permutations with repetition nr For n elements with n1, n2 … nt identical objects For n unique elements in X with r selections from X permutations with repetition of identical objects n! n1!n2! …nt! combinations with repetition C(n+r-1,r)