Homework Problems for
Permutations & Combinations
Homework Solutions
Find the following:
1. P(7,5)
2. C(7,5)
3. P(4,4)
4. P(n,n)
5. 9!
6. C(5,5)
7. C(n,n)
8. C(8,0)
9. C(n,0)
10. P(12,3)
11. C(9,2)
12. P(18,8)
13. C(21,17)
14. P(7,4)
15. C(11,1)
16. P(9,0)
17. C(9,0)
18. P(8,3)
19. C(13,5)
20. Twenty applicants for a secretary position are to be interviewed to narrow the list of candidates to the top five. How many possible results are there if
A) the top five are ranked in order of preference?
B) the top five are unranked?
21. Six speakers are scheduled to address a group of College students. In how many different orders can the speakers appear?
22. In how many different ways can the letters of the work HOUSE be arranged?
23. A company has divided a state into eight regions. It wishes to test a product in three of these regions. How many different ways are there to select these three regions?
24. The chocolate factory classifies its candies as caramels (10 types), chocolate (7 types), and dark chocolate (8 types). A customer has ordered an assortment to consist of six types of caramels, four types of chocolate, and five types of dark chocolates. How many such assortments are possible?
25. A five member committee is to be selected from among four Math teachers and five English teachers. In how many different ways can the committee be formed under the following circumstance?
A) Anyone is eligible to serve on the committee.
B) The committee must consist of 3 Math teachers and 2 English teachers.
C) The committee must contain at least three Math teachers.
D) The committee must contain at least three English teachers.
26. From a group of 8 teachers, a committee of at least one and at most three persons is to be formed. How many different committees can be formed?
27. In Rapid City South Dakota, there are 10 dogs racing for first and second prize.
How many possible outcomes are there?
28. If there are 8 orange bars, 9 red bars and 5 blue bars, how many different ways are there to give a person 2 orange bars, 3 red bars and 1 blue bar?
29. How many different ways are there to draw 6 cards from a standard deck of cards and obtain 4 kings and 2 jacks?
30. How many different ways are there to draw 7 cards from a standard deck of cards and obtain 3 jacks and 4 queens?
31. If two cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the two cards if at least one card is a jack, queen or a king?
32. If three cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the three cards if at least two cards are a jack, queen or a king?
33. How many different ways can the letters in the word crystal be arranged?
34.
There are 23 male and 25 female students at West Mount High.
Two different scholarships are to be awarded.
How many ways are there to award one of the scholarships to a male student and the other to a female student.
35.
There are 23 male and 25 female students at West Mount High.
Two scholarships which are identical are to be awarded.
How many ways are there to award one of the scholarships to a male student and the other to a female student.
1. 2520
2. 21
3. 4! = 24
4. n!
5. 362880
6. 1
7. 1
8. 1
9. 1
10. 1320
11. 36
12. 1,764,322,560
13. 5985
14. 840
15. 11
16. 1
17. 1
18. 336
19. 1287
20. 1,860,480, 15,504
21. 720
22. 120
23. 56
24. 411,600
25. 126, 40, 45, 81
26. 92
27. P(10,2) = 90
28. 11,760
29. C(4,4)*C(4,2) = (1)(6) = 6
31.
Let the set of jack, queen, king be called set A. There are 12 cards in this set.
Let the other cards be set B. There are 40 cards in this set.
Draw two cards. S = {1 card from set A and 1 card from set B, or both cards from set A, or ...
S = {AB, AA, BB}
We want at least one from set A, so find AB or AA.
C(40,1)*C(12,1) + C(12,2) = 546
32. 2860
33. 7! = 5040
Or P(7,7)
34. 2(23)(25) = 1150
35. (23)(25) = 575