Combinatorics - MATH 0345
Exam 2
November 17, 2006
Name:
Honor Code Pledge
Signature
Directions: Please complete all but 1 problem. If you complete all six problems, I
will count your best five.
1. Let nand kbe positive integers. Give a combinatorial proof that
Σn
k=1kn
k2
=n2n−1
n−1.
2. Given that 1/2
k=(−1)k−1
k22k−12k−2
k−1, use Newton’s Binomial Theorem to approximate
√40. (You may leave your answer as a sum.)
3. An opinion poll reports that the percentage of voters who would be satisfied with
each of three candidates A, B, C for President is 65%,57%,58% respectively. Further,
28% would accept Aor B, 30% Aor C, 27% Bor Cand 12% would be content with
any of the three. What do you conclude?
4. What is the number of ways to place six nonattacking rooks on the 6-by-6 boards
with forbidden positions as shown? (See opposite side. You may leave your answer
as a sum.)
5. Prove that the nth Fibonacci number fnis the integer that is closest to the number
1
√5(1 + √5
2)n.
6. •There are nseating positions arranged in a line. Prove that the number of ways
of choosing a subset of these positions, with no two chosen positions consecutive,
is fn+1 (the (n+ 1)th fibonacci number).
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