Module 7 Complexity Module 151 Combinatorics Rosen 5

Module #7 - Complexity Module #15_1: Combinatorics Rosen 5 th ed. , §§ 4. 4 -4. 5 ~12 slides, ~2 lecture 2021/9/8 (c)2001 -2003, Michael P. Frank 1

Module #7 - Complexity Binomial Coefficients • • 2021/9/8 (c)2001 -2003, Michael P. Frank 2

Module #7 - Complexity Examples • E. g. 4: What is the coefficient of x 12 y 13 in the expansion of (2 x-3 y)25? • Corollary 1: • Corollary 2: • Corollary 3: 2021/9/8 (c)2001 -2003, Michael P. Frank 3

Module #7 - Complexity Pascal’s Identity and Triangle • Pascal’s identity: • Pascal’s Triangle: Figure 1 (pp. 331) 2021/9/8 (c)2001 -2003, Michael P. Frank 4

Module #7 - Complexity Vandermonde’s Identity • Vandermonde’s identity: • Proof: Suppose there are m items in one set and n items in a 2 nd set. The proof follows that if we take r items from these 2 sets. • Corollary 4: 2021/9/8 (c)2001 -2003, Michael P. Frank 5

Module #7 - Complexity Vandermonde’s Identity • Theorem 4: • Proof: L. H. S. =counts bit strings of length n+1 containing r+1 ones. Consider the possible locations of the final 1 in a string with r+1 ones. The final 1 must at position r+1, r+2, … or n+1. If the last one is at k-th position, there must be r ones among the first k-1 positions. Consequently, there are C(k-1, r) such bit strings. Summing over k with r+1≦k≦n+1, we have bit strings of length n containing exactly r+1 ones. 2021/9/8 (c)2001 -2003, Michael P. Frank 6

Module #7 - Complexity Permutations with Repetition • Theorrem 1: The number of r-permutations of a set of n objects with repetition allowed is nr. • E. g. How many strings of length n can be formed from the English alphabet. 2021/9/8 (c)2001 -2003, Michael P. Frank 7

Module #7 - Complexity Combinations with Repetition • E. g. 2: How many ways to choose 4 pieces of fruit with all types from a bowl containing apples, oranges, and pears? • E. g. 3: How many ways to select five bills from a cash box containing $1, $2, $5, $10, $20, $50, and $100 bills? 2021/9/8 (c)2001 -2003, Michael P. Frank 8

Module #7 - Complexity Combinations with Repetition • Theorem 2: There are C(n+r-1, r) rcombinations from a set with n elements when repetition of elements is allowed. • Proof: similar to e. g. 3 2021/9/8 (c)2001 -2003, Michael P. Frank 9

Module #7 - Complexity Combinations with Repetition • • 2021/9/8 E. g. 4 E. g. 5 E. g. 6 E. g. 7 (c)2001 -2003, Michael P. Frank 10

Module #7 - Complexity Permutations with Indistinguishable Objects • How many different strings can be made by reordering “SUCCESS”? Ans: C(7, 3)C(4, 2)C(2, 1)C(1, 1)=7!/(3!2!1!1!)=420 Theorem 3: The number of different permutations of n objects, where there are n 1 indistinguishable objects of type 1, n 2 indistinguishable objects of type 2, …, nk indistinguishable objects of type k, is 2021/9/8 (c)2001 -2003, Michael P. Frank 11

Module #7 - Complexity Distributing Objects into Boxes • E. g. 9: How many ways to distribute hands of 5 cards to each of four players from the standard deck of 52 cards? Ans: C(52, 5)C(47, 5)C(42, 5)C(37, 5)= • Theorem 4: The number of ways to distribute n distinguishable objects into k distinguishable boxes so that ni objects are placed into box I, i=1, 2, …, k, equals 2021/9/8 (c)2001 -2003, Michael P. Frank 12