Unformatted text preview: the committee must contain more students than faculty? Binomial Theorem and Combinatorial Proofs 1. Show that n X k =0 2 k ± n k ² = 3 n 2. What is the coeﬃcient of x 3 y 5 in the expainsion of (2 x + 3 y ) 8 . Hint: be careful. .. 3. Prove that ( 2 n 2 ) = 2 ( n 2 ) + n 2 . Hint: for the RHS, break it down into cases of where the 2 elements come from. 4. A rectangular city’s roads are laid out like a grid in which (0,0) is the most southwest corner of the city and ( m,n ) is the most northeast. If there’s a road at every integer (both north-south and east-west), how many ways can you get from (0,0) to (m,n) assuming you never backtrack (that is, you never go south or west)? 5. Prove that ( n r )( r k ) = ( n k )( n-k r-k ) by using a combinatorial argument...
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- Summer '08
- Permutations, Playing card, Combinatorial proofs, Rob Bayer, Rob Bayer Permutations