Unformatted text preview: P . (4) A pile oF n identical coins are to be given to Adam, Brenda, Clarisse, Darryl, and Ethel; Adam and Brenda must each receive at least 2 coins, and Ethel can receive no more than 2. Let a n be the number oF ways to distribute all n oF the coins. ²ind a Formula For a n . (5) Let X be a set and Y ⊆ X such that  X  = n and  Y  = k . ²ind a closed Formula For the number oF surjective Functions X → Y (and prove its correctness). (6) Let a n be the number oF subsets oF { 1 , . . ., n } with no two consecutive elements. ²ind a closed Formula For a n . (7) Let X be a fnite set, and let Π 1 and Π 2 two partitions oF X such that  Π 1  = n and  Π 2  = n + 1. We call an element oF x ∈ X declining , iF the part oF Π 2 that contains x has Fewer elements than the part oF Π 1 that contains x . Prove that X has at least two declining elements. Date : November 28, 2011. 1...
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 Fall '09
 WILDSTROM
 Combinatorics, Set Theory, Binomial, Adam, maximal chains, COMBINATORICS MASTERS EXAM

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