1) Use a flagpole model to prove that n + 1 3 = Xn1 k=1 k(n k).

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\fA donut shop sells 4 kinds of donuts: chocolate, glazed, jelly and sugar. How many different ways are there to buy 10 donuts a) with no restrictions? b) including at least 2 jelly and 3 glazed? 0) including no more than 5 chocolate? There are 7 people (3 men and 4 women) at a dinner party. In how many ways can they be seated around a circular table if a) there are no restrictions? b) no two men can be seated side by side? (3) Eric and Jean insist on sitting next to each other? 10. How many paths, always moving right or upward, join A to B in the following street patterns: a) B(10, 6) b) .B(9, 6) i i A(0, 0) A(0, 0) Hint: All paths must go through Hint: Compared to a complete grid, the point (6, 4) how many paths have been removed? 11. How many permutations of the multiset {314, 53,80} are there with: a) no restrictions? b) no adjacent As