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hw3sol

# hw3sol - MATH 55 Fall 2007 Prof Bernoff Harvey Mudd College...

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MATH 55 Prof. Bernoff Fall 2007 Harvey Mudd College Homework 3 – Solutions 17.6 Prove: n k = k + 1 n - 1 . Solution: There is a bijective argument that these two are equal using the idea of two arrangements that are duals - that is two arrangements that can be made into each other by exchanging how objects are labelled. The number of ways to arrange k bars and n - 1 stars is the same way as the number of ways to arrange k stars and n - 1 bars. By the stars and bars argument, (( n k )) counts the number of ways to place k stars around n - 1 bars. Similarly, (( k +1 n - 1 )) counts the number of ways to place n - 1 stars around k bars, and so they’re equal. 17.7 Let n k denote the number of multisets of cardinality k we can form choosing the elements in { 1 , 2 , 3 , ..., n } with the added condition that we must use each of these n elements at least once in the multiset. (a) Evaluate from first principles, n n . Solution: There is only one way to pick a set of n elements from { 1 , 2 , 3 , ..., n } with each number represented at least once. (b) Prove: n k = ( n k - n ) . Solution: After choosing each of 1 , . . . , n for a multiset of size k , we choose the remaining k - n elements as a multiset from { 1 , 2 , 3 , ..., n } , so n k = (( n k - n )) . 17.8 Let n, k be positive integers. Prove: n k = n - 1 0 + n - 1 1 + n - 1 2 · · · + n - 1 k Solution: There are (( n k )) k -multisets of { 1 , 2 , 3 , ..., n } , but we can count them all by conditioning on how many times n appears in each multiset. The k -multisets in which n appears i times are the (( n - 1 k - i )) ( k - i )-multisets of { 1 , 2 , 3 , ..., n - 1 } (which fill in the rest of each k -multiset). Since i can range from 0 to k , we have (( n k )) = k i =0 (( n - 1 k - i )) . 17.9 Let n, k be positive integers. Prove: n k = 1 k - 1 + 2 k - 1 + 3 k - 1 · · · + n k - 1 Solution: We count the k -multisets of { 1 , 2 , 3 , ..., n } by conditioning on the largest element of each. If i is the largest element in a k -multiset of { 1 , 2 , 3 , ..., n } , there are (( i k - 1 )) ways to choose the remaining k - 1 elements for the multiset. Since

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hw3sol - MATH 55 Fall 2007 Prof Bernoff Harvey Mudd College...

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