(1) Let
k be any positive integer. Prove that there exists a positive
integer multiple n of k such that the only digits in n are 0s and
1s. (Use the pigeonhole principle.)
(2) A ternary string is a sequence of 0s, 1s and 2s. How many
ternary strings of length 12 are there? How many of those
strings contain exactly five 0s, three 1s and four 2s? How many
ternary strings of length 12 contain an odd number of 1s?
(3) Prove the binomial identity
X
n
i=0
n
i
2
=
2
n
n
.
(4) A committee of seven people is to be chosen from twelve married
couples. How many ways can a committee of three men and four
women be chosen? How many ways can a committee containing
no married couple be chosen? How many ways can a committee
containing exactly two married couples be chosen?
(5) Find the number of non-negative integer solutions of
x
1 + x2 + x3 + x4 + x5 = 7.
How many of those solutions have xi 0 for all i?