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master - P(4 A pile oF n identical coins are to be given to...

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COMBINATORICS MASTERS EXAM Instructions: Work out as many problems as you can. Focus on com- plete and correct solutions, because partial credit will only be sparingly awarded, and complete solutions are required to pass. Binomial coef- cients, Stirling numbers, ±actorials are allowed in closed ±ormulas, but no , p or . . . . (1) A fnished tic-tac-toe board is a 3 × 3 square grid with 5 X’s and 4 O’s in the squares. Compute the number oF tic-tac-toe boards where two boards count as identical, iF one can be transFormed into the other by re±ection or rotation oF the board. (2) ²ind the numbers oF n -letter strings consisting oF As, Bs, Cs, and Ds with at least one A, an even number oF Bs, an odd number oF Cs, and any number oF Ds. (3) Let P be the poset on the subsets oF the set { 1 , . . ., 12 } , ordered by inclu- sion. (a) ²ind the number oF elements oF P . (b) ²ind the number oF maximal chains P . (c) ²ind the number oF maximal chains oF P that go through { 1 , 5 , 6 , 7 , 11 } . (d) ²ind the width oF
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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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