MATH 145: Homework 1
Due: Wednesday, August 10th, in class
Remember that you need to show your work and explain how you got your answer to receive credit.
1. What is the number of subsets of a set with n elements, containing a given element?
2. In a sandwich shop there are 5 different kinds of bread, 4 different kinds of cheese, and 3 different kinds of
meat. How many different sandwiches can be constructed (assume that each sandwich must have exactly one
type of bread, cheese, and meat)?
3. We have 20 different presents that we want to distribute to 12 children. It is not required that every child get
something; it could even happen that we give all the presents to the same child. In how many ways can we
distribute the presents?
4. Compute all values of nk for 0 ≤ k ≤ n ≤ 5.
5. Recall that we define the “degree” of the polynomial
f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0
(1)
to be n. How many degree 4 polynomials are there whose coefficients belong to the set {0, 1, 2, 3}?
6. Find all positive integers a, b, and c for which
a b
a
=2
.
b
c
c
(2)
7. Find the number of all 20-digit integers in which no two consecutive digits are the same.
8. Let B be a subset of A, |A| = n, |B| = k. What is the number of all subsets of A whose intersection with B
has 1 element?
9. Prove by induction that the sum of the first n positive integers is n(n + 1)/2.
10. Use induction to prove that the number of permutations of n objects is n!.
11. Use induction to prove that the sum of the first n powers of 2 (starting with 1 = 20 ) is 2n − 1.
12. There is a class of 40 girls. There are 18 girls who like to play chess, and 23 who like to play soccer. Several of
them like biking. The number of those who like to play both chess and soccer is 9. There are 7 girls who like
chess and biking, and 12 who like soccer and biking. There are 4 girls who like all three activities. In addition
we know that everybody likes at least one of these activities. How many girls like biking?
13. We select 38 even positive integers, all less than 1000. Prove that there will be two of them whose difference
is at most 26.
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