Discrete Mathematics
Problem Sheet 2
Enumerations of Combinations & Permutations:
1. Find the number of ways in which 5 children can ride a toboggan if 1 of
the 3 oldest children must drive?
2. A group of 8 scientists is composed of 5 psychologists and 3 sociologists.
(a) In how many ways can a committee of 5 be formed?
(b) In how many ways can a committee of 5 be formed that has 3 psy-
chologists and 2 sociologists?
3. How many 4-digit telephone numbers have one or more repeated digits?
4. (a) How many binary sequences are there of length 15?
(b) How many binary sequences are there of length 15 with exactly six
1’s?
5. A farmer buys 3 cows, 8 pigs, and 12 chickens from a man who has 9 cows,
25 pigs, and 100 chickens. How many choices does the farmer have?
6. Find the number of ways in which 5 different English books, 6 French
books, 3 German books, and 7 Russian books can be arranged on a shelf
so that all books of the same language are together.
7. How many 9-letter words can be formed that contain 3,4, or 5 vowels,
(a) Allowing repetition of letters?
(b) Now allowing repetition?
8. How many ways can 5 days be chosen from each of the 12 months of an
ordinary year of 365 days/
9. A bag contains 20 distinguishable balls of which 6 are red, 6 are white,
and 8 are blue. We draw out 5 balls with at least one red ball, replace
them, and then draw 5 balls with at most one white one. How many ways
can this be done?
10. How many ways can 3 integers be selected from the integers 1,2, . . . , 30
so that their sum is even?
11. How many ways can a person invite 3 of his 6 friends to lunch every day
for 20 days?
Permutations & combinations with repetitions:
1. A quarterback of a football team has a repertoire of 20 plays and runs 60
plays in the course of a game. The coach is interested in the frequency
distribution of the play-calling showing how many times each of the various
plays were called. How many such frequency distribution are there?
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