Unformatted text preview: MATH 681 Problem Set #1 Show work for each problem. Answers without justiﬁcation, or justiﬁed solely by direct
enumeration, are not acceptable. Arithmetic expressions may be left unsimpliﬁed (e.g. if
you can explain why an answer is computable as being 67 − 5 · 23 , that is more than suﬃcient;
it is not necessary to then calculate that 67 − 5 · 23 is 279896).
This problem set is due at the beginning of class on September 3.
1. (10 points) A ﬁve-digit number must not have a zero as its ﬁrst digit.
(a) (5 points) How many ﬁve-digit numbers are there whose digits appear in increasing order (e.g., 25689)?
(b) (5 points) How many ﬁve-digit numbers are there whose digits appear in descending order (e.g., 98652)?
2. (10 points) The members of two committees are being drawn from a pool of eight
people. Each person can be on either committee or on no committee at all.
(a) (5 points) How many ways are there to assign committee-responsibilities to the
eight people if there are no restrictions on committee membership?
(b) (5 points) How many ways are there to assign committee-responsibilities to the
eight people if each committee must have at least one member?
3. (10 points) There are two straightforward ways to prove that 2n
2 =2 n
2 + n2 . (a) (5 points) Prove it algebraically, by manipulating the two sides to get equal
(b) (5 points) Prove it combinatorially, by ﬁnding a set counted by the left side and
demonstrating that the right side counts the same set.
4. (10 points) Prove the following identity by demonstrating that the two sides of the
equation are counting the same thing:
n k=0 n
n−k Brute force is the last refuge of the incompetent. Page 1 of 1 = 2n
n —Traditional hacker lore Due September 3, 2009 ...
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- Fall '09