MATH 681 Exam #1 Answer exactly four of the following six questions. Indicate which four you would like graded! Binomial coeﬃcients, Stirling numbers, and arithmetic expressions need not be simpliﬁed in your answers. 1. (10 points) Prove the combinatorial identity ∑ n k =1 k 2 ( n k ) = n 2 n-1 + n ( n-1)2 n-2 . You may use any method you like. 2. (10 points) Answer the following questions relating to paths through the following grid (note the excluded portions): (a) (5 points) How many walks are there from the lower left corner to the upper right corner taking upwards and rightwards steps only? (b) (5 points) How many of these walks pass through the point marked with a solid dot? 3. (10 points) Answer the following questions about the number of solutions a n to the equation x 1 + x 2 + x 3 + x 4 = n subject to the conditions that all the x i are non-negative integers, that x 1 ≤ 3, 2 ≤ x 2 ≤ 4, x 3 ≥ 5, and x 4 ≥ 5. (a)
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