288
Sets, Combinatorics, and Probability
exeRciSeS 4.4
1. Compute the value of the following expressions.
a. P(7, 2)
b. P(8, 5)
2. Compute the value of the following expressions.
a. P(6, 4)
b. P(n, n − 1)
3. How many batting orders are possible for a 9-man baseball team?
4. The 14 teams in the local Little League are listed in the newspaper. How many listings are possible?
5. How many different ways can 10 flavors of ice cream be arranged in an ice cream store display case?
6. How many different ways are there to arrange 6 candidate names on a ballot?
7. How many permutations of the characters in COMPUTER are there? How many of the permutations end
in a vowel?
8. In how many ways can 6 people be seated in a circle of 6 chairs? Only relative positions in the circle can
be distinguished.
9. In how many ways can first, second, and third prize in a pie-baking contest be given to 15 contestants?
10. a. Stock designations on an exchange are limited to 3 letters. How many different designations are there?
b. How many different designations are there if letters cannot be repeated?
11. In how many different ways can 19 people be seated in a row?
12. In how many different ways can 11 men and 8 women be seated in a row?
13. In how many different ways can 11 men and 8 women be seated in a row if the men all sit together and the
women all sit together?
14. In how many different ways can 11 men and 8 women be seated in a row if no 2 women are to sit together?
15. In how many different ways can 11 men and 8 women be seated around a circular table? (Only relative
positions in the circle can be distinguished.)
16. In how many different ways can 11 men and 8 women be seated around a circular table if no 2 women are
to sit together? (Only relative positions in the circle can be distinguished.)
S e c t i o n 4 . 4 review
tecHniQueS
• Find the number of permutations of r distinct
objects chosen from n distinct objects.
• Find the number of combinations of r distinct
objects chosen from n distinct objects.
• Use permutations and combinations in conjunction
with the multiplication principle and the addition
principle.
• Find the number of distinct permutations of n
objects that are not all distinct.
• Find the number of permutations of r objects
out of n distinct objects when objects may be
repeated.
• Find the number of combinations of r objects out
of n distinct objects when objects may be repeated.
• Generate all permutations of the integers {1, … , n}
in lexicographical order.
• Generate all combinations of r integers from the set
{1, … , n}.
main iDeaS
• There are formulas for counting various permutations and combinations of objects.
• Care must be taken when analyzing a counting
problem to avoid counting the same thing more
than once or not counting some things at all.
• Algorithms exist to generate all permutations of
n objects and all combinations of r out of n objects.
W
Sets, Combinatorics, and Probability
exeRciSeS 4.4
1. Compute the value of the following expressions.
a. P(7, 2)
b. P(8, 5)
2. Compute the value of the following expressions.
a. P(6, 4)
b. P(n, n − 1)
3. How many batting orders are possible for a 9-man baseball team?
4. The 14 teams in the local Little League are listed in the newspaper. How many listings are possible?
5. How many different ways can 10 flavors of ice cream be arranged in an ice cream store display case?
6. How many different ways are there to arrange 6 candidate names on a ballot?
7. How many permutations of the characters in COMPUTER are there? How many of the permutations end
in a vowel?
8. In how many ways can 6 people be seated in a circle of 6 chairs? Only relative positions in the circle can
be distinguished.
9. In how many ways can first, second, and third prize in a pie-baking contest be given to 15 contestants?
10. a. Stock designations on an exchange are limited to 3 letters. How many different designations are there?
b. How many different designations are there if letters cannot be repeated?
11. In how many different ways can 19 people be seated in a row?
12. In how many different ways can 11 men and 8 women be seated in a row?
13. In how many different ways can 11 men and 8 women be seated in a row if the men all sit together and the
women all sit together?
14. In how many different ways can 11 men and 8 women be seated in a row if no 2 women are to sit together?
15. In how many different ways can 11 men and 8 women be seated around a circular table? (Only relative
positions in the circle can be distinguished.)
16. In how many different ways can 11 men and 8 women be seated around a circular table if no 2 women are
to sit together? (Only relative positions in the circle can be distinguished.)
S e c t i o n 4 . 4 review
tecHniQueS
• Find the number of permutations of r distinct
objects chosen from n distinct objects.
• Find the number of combinations of r distinct
objects chosen from n distinct objects.
• Use permutations and combinations in conjunction
with the multiplication principle and the addition
principle.
• Find the number of distinct permutations of n
objects that are not all distinct.
• Find the number of permutations of r objects
out of n distinct objects when objects may be
repeated.
• Find the number of combinations of r objects out
of n distinct objects when objects may be repeated.
• Generate all permutations of the integers {1, … , n}
in lexicographical order.
• Generate all combinations of r integers from the set
{1, … , n}.
main iDeaS
• There are formulas for counting various permutations and combinations of objects.
• Care must be taken when analyzing a counting
problem to avoid counting the same thing more
than once or not counting some things at all.
• Algorithms exist to generate all permutations of
n objects and all combinations of r out of n objects.
W