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Unformatted text preview: MATH 681 Problem Set #2 Show work for each problem. Answers without justiﬁcation, or justiﬁed solely by direct
enumeration, are not acceptable. Arithmetic expressions may be left unsimpliﬁed.
This problem set is due at the beginning of class on September 17.
1. (10 points) Prove the following identity combinatorially:
n i
i=0 n
i = n2n−1 2. (10 points) If a fair coin is ﬂipped n times, what is the probability that
(a) (3 points) The ﬁrst head comes after exactly m tails?
(b) (7 points) The ith head comes after a total of m previous tails?
3. (10 points) Using paths through a lattice (or some other combinatorial object, if you
prefer), prove that the following identity is true for any m ≤ k ≤ n:
m+n
n m =
i=0 k
i m+n−k
m−i 4. (10 points) Construct generating functions for the number of nonnegative integer
solutions to the following equations:
(a) (5 points) x1 + x2 + x3 = n for x1 ≥ 3, x2 ≤ 4, and 2 ≤ x3 ≤ 5.
(b) (5 points) x1 + 2x2 + 5x3 = n for x3 ≤ 2 (and no restrictions on the other two).
5. (4 point bonus) It follows from the binomial theorem that
0. Can you ﬁnd a combinatorial proof of this identity? n
nn
i=1 (−1)
i = (1−1)n = From two letters or forms are composed two dwellings; from three, six; from four,
twentyfour; from ﬁve, one hundred and twenty; from six, seven hundred and twenty;
from seven, ﬁve thousand and forty; and thence their numbers increase in a manner
beyond counting and are incomprehensible.
—Sefer Yetzira Page 1 of 1 Due September 17, 2009 ...
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.
 Fall '09
 WILDSTROM
 Math

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