1. a. Suppose I have 3 history books, 4 geography books, and 5 English
books. How many ways can I arrange the books on a shelf if they are grouped
by subject?
b. How many ways can I distribute the books among
6 people, if no one gets more than one book on a particular subject?
2. How many ways can 15 identical candy bars be distributed among 6 people?
3. How many distinct 11-character strings can be made out of the letters in SALESPERSON?
4. What is the probability of being dealt a 5-card hand containing three cards of the same denomination, with no other matching cards?
5. What is the probability of being dealt a 13-card hand that contains no card higher than a 10? Count the ace as a high card.
6. [Place] Let S = {1, 2, ... 12}. Suppose we choose 7 distinct numbers from S. Prove that at least 2 of the numbers chosen must sum to 13.
7. Given: x + y + z = 15. How many non-negative integer solutions exist for x, y, z?
8. Suppose we have a bowl full of marbles: 6 red, 9 green, 4 blue, 6
clear.
a. How many ways can we draw (3 red) and (5 green)
in 8 draws, assuming no replacement?
b. How many ways can we draw (3 red) and (at least
3 green) in 8 draws, assuming no replacement?
c. How many ways can we draw (3 green) and (5 clear)
in 8 draws, assuming marbles are replaced?
9. Suppose we flip a coin 10 times. Each flip has a single outcome of
either heads or tails.
a. What is the probability that we will obtain exactly
five heads?
b. What is the probability of getting 5 consecutive
heads in 10 flips, with the rest of the flips being tails?
c. What is the probability of getting 7 or more
heads?
d. What is the probability of getting 10 heads?
10. Take the following as given:
i..) There are more than six billion people
in the world.
ii.) A human head contains fewer than 200,000
hairs.
a. Prove that there are at least two people with
exactly the same number of hairs on their head.
b. Can we claim that for some integer n,
there are at least 30,000 people with that number of hairs on their head?
Why or why not?
c. Can we claim that for some integer n, there
are at least 1,000,000 people with that number of hairs on their head?
Why or why not? If not, can we claim that such a number does not
exist, or only that its existence cannot be proven?
11. A drawer contains 50 white socks and 50 black socks. How many draws are necessary to ensure we have a matching pair? How many draws are necessary to ensure that we have a pair of black socks?
12. A couple has 2 children. One of the children is a boy; no information is known about the other. I claim that the probability that the couple has 1 boy and 1 girl is not 1/2, but rather is 2/3. Prove or disprove the validity of this statement.
13. Twenty-four people are to be seated around a circular table. There is a place card in front of each seat, designating who is to sit in each seat. The guests sit down, each selecting a seat at random. Checking the cards, it is found that no one is sitting in the seat reserved for them. Prove or disprove that it is possible to move everyone n seats in the same direction to ensure that at least two people are sitting their correct seats. [Taken from Mathematical Circus, by Martin Garder.]
14. Another from Gardner: A standard 52-card deck is thoroughly shuffled.
The top card is removed and its color noted, and it is then replaced on
top of the deck. The deck is then cut--that is, some group of cards is
removed from the top of the deck, and moved as a group to the bottom. The
card just moved to the top of the deck is removed and examined. What is
the probability that this card is the same color as the first card that
was examined?