MATH 310 Elem Combinatorics Penn State

Brualdi, Fourth Edition, Chapter 5, Problem 11 Consider the set of elements S = {1, 2, 3, 4, . . . , n}. Then we know that n counts the number of k-element k subsets of S, while n-3 counts the number of k-element subsets of S which DO NOT have 1, 2,

Brualdi, Fourth Edition, Chapter 5, Problem 12 a) The fact that this sum equals 0 when n is odd comes from the symmetry of Pascals triangle; that is, the n fact that n = nk for all 0 k n. Hence, we have k n (1)k k=0 n k 2 = n 0 2 2 n 1 2

Brualdi, Fourth Edition, Chapter 1, Problem 1 First, assume at least one of m or n is even. (Without loss of generality, assume m is even.) Then the number of rows of the board is even, say m = 2k. That means that k dominoes can be placed in each col

Brualdi, Second Edition Chapter 5, Problem 11 Use combinatorial reasoning to prove that n n-3 - k k = n-1 n-2 n-3 + + . k-1 k-1 k-1 Solution Consider the set of elements S = {1, 2, 3, 4, . . . , n}. Then we know that n counts the number of k-element

Brualdi, Second Edition Chapter 5, Problem 12 Prove that n (-1)k k=0 n k 2 () a) equals 0 if n is odd and b) equals (-1)m Solution a) The fact that () equals 0 when n is odd comes from the symmetry of Pascal\'s triangle; that is, the fact n that

MATH 310 Sample Questions Project Spring 2007 Name: James Sellers Name: Mary Sellers Question 1 (from Section 3.2) How many ways can .? a) 2 b) 5 c) 10! d) 27 * Question 2 (from Section 5.1) What is .? a) C(7,2) b) C(5,4) c) C(10, 9

Brualdi, Fourth Edition, Chapter 5, Problem 11 Consider the set of elements S = {1, 2, 3, 4, . . . , n}. Then we know that n counts the number of k-element k subsets of S, while n-3 counts the number of k-element subsets of S which DO NOT have 1, 2,

Brualdi, Fourth Edition, Chapter 5, Problem 12 a) The fact that this sum equals 0 when n is odd comes from the symmetry of Pascals triangle; that is, the n fact that n = nk for all 0 k n. Hence, we have k n (1)k k=0 n k 2 = n 0 2 2 n 1 2

MATH 310 Sample Questions Project Spring 2007 Name: James Sellers Name: Mary Sellers Question 1 (from Section 3.2) How many ways can .? a) 2 b) 5 c) 10! d) 27 * Question 2 (from Section 5.1) What is .? a) C(7,2) b) C(5,4) c) C(10, 9