Assignment 1 Soln

# Assignment 1 Soln - MATH 239 Assignment 1 Solutions This...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 239 Assignment 1 Solutions This assignment is due at noon on Friday, May 15, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Let integers 0 ≤ ` ≤ k ≤ n be given. Consider the binomial identity n k k ` = n ` n- ` k- ` . (a) Give an algebraic proof of this identity. Solution: Using the formula ( n k ) = n ! ( n- k )! k ! we find LHS = n ! ( n- k )! k ! k ! ( k- ` )! ` ! = n ! ( n- k )!( k- ` )! ` ! and RHS = n ! ( n- ` )! ` ! ( n- ` )! ( n- k )!( k- ` )! = n ! ( n- k )!( k- ` )! ` ! , so LHS=RHS as required. (b) Give a combinatorial proof of this identity. Solution: We define a set S of pairs of subsets as follows: S = { ( A, B ) : A ⊆ B ⊆ { 1 , 2 , . . . n } , | A | = `, | B | = k } . We calculate the size of the set S in two different ways. First, we choose k elements of { 1 , 2 , . . . , n } in all possible ways to form the set B , then from the k elements of B we choose ` elements in all possible ways to form the set A . Therefore the number of elements of S is ( n k )( k ` ) . Now we calculate the size of S by choosing the set A first. We can do this by choosing ` elements of { 1 , 2 , . . . , n } in all possible ways. Then to form the set B , we must choose k- ` elements from the remaining n- ` elements of { 1 , 2 , . . . , n } that were not chosen to be elements of A . There are therefore ( n- ` k- ` ) such choices. Therefore the size of S is ( n ` )( n- ` k- ` ) . Thus we have ( n k )( k ` ) = | S | = ( n ` )( n- ` k- ` ) as required....
View Full Document

## This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

### Page1 / 4

Assignment 1 Soln - MATH 239 Assignment 1 Solutions This...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online