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# ws717 - the committee must contain more students than...

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Rob Bayer Math 55 Worksheet July 17, 2009 Permutations, Combinations 1. How many bit strings of length n (a) contain exactly four 1’s? (b) contain at least two 0’s? (c) contain the same number of 0’s and 1’s? Assume n is even. 2. Recall that a poker hand consists of 5 cards from a standard 52-card deck. How many ways are there to get a: (a) four-of-a-kind? (b) two-of-a-kind? (c) hand with no pairs? (d) flush? (All cards the same suit) (e) straight? (A straight is five cards with consecutive ranks, but can be of any suit) (f) straight flush? (combine the above two) 3. How many odd-sized subsets are there of a set with 10 elements? With n ? Your answer should be a closed form. 4. Show that if p is prime and 1 k < p , then ( p k ) is divisible by p 5. How many different 5-person committees can be made if there are 100 students and 30 faculty members, and
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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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