Section 4.4 Permutations and Combinations
289
17. Compute the value of the following expressions.
a. C(10, 7)
b. C(9, 2)
c. C(8, 6)
18. Compute C(n, n − 1). Explain why C(n, n − 1) = C(n, 1).
19. Quality control wants to test 25 microprocessor chips from the 300 manufactured each day. How many
different batches of test chips are possible?
20. A soccer team carries 18 players on the roster; 11 players make a team. How many different teams are
possible?
21. How many juries of 5 men and 7 women can be formed from a panel of 17 men and 23 women?
22. How many different sets of 4 novels and 3 plays can be created from a collection of 21 novels and
11 plays?
Exercises 23–26 deal with the following situation: Of a company’s personnel, 7 people work in design, 14 in
manufacturing, 4 in testing, 5 in sales, 2 in accounting, and 3 in marketing. A committee of 6 people is to be
formed to meet with upper management.
23. How many committees with 1 member from each department are possible?
24. How many committees with exactly 2 members from manufacturing are possible?
25. How many committees with no representative from accounting and exactly 1 representative from marketing are possible?
26. How many committees with at least 2 representatives from manufacturing are possible?
Exercises 27–32 concern a 5-card hand from a standard 52-card deck. A standard deck has 13 cards from each
of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through10, jack, queen, king, or ace.
Each face value is a “kind” of card. The jack, queen, and king are “face cards.”
27. How many hands contain 4 queens?
28. How many hands contain all diamonds?
29. How many hands contain 3 spades and 2 hearts?
30. How many hands contain cards from all 4 suits?
31. How many hands consist of all face cards?
32. How many hands contain exactly 2 spades and exactly 2 hearts?
Exercises 33–42 concern 5-card poker hands from a standard 52-card deck.
33. How many hands contain a royal straight flush (that is, the 10, jack, queen, king, ace of one suit)?
34. How many hands contain a straight flush (that is, 5 consecutive cards of the same suit, where aces can be
low or high) that is not a royal straight flush (see Exercise 33)?
35. How many hands contain four of a kind (such as 4 jacks plus a fifth card)?
36. How many hands contain a full house (that is, three of a kind plus a pair of another kind)?
37. How many hands contain a flush (that is, 5 cards of the same suit) that is not a straight flush or a royal
straight flush (see Exercises 33 and 34)?
38. How many hands contain a straight (that is, 5 consecutive cards, where aces can be low or high) that is not
a straight flush or a royal straight flush (see Exercises 33 and 34)?
39. How many hands contain three of a kind (that is, exactly 3 cards of the same kind plus 2 other cards that
are not a pair)?
40. How many hands contain 2 pairs (that is, 2 pairs of 2 different kinds plus a fifth card of some third kind)?
41. How many hands contain 1 pair (that is, exactly 2 cards of the same kind)?
289
17. Compute the value of the following expressions.
a. C(10, 7)
b. C(9, 2)
c. C(8, 6)
18. Compute C(n, n − 1). Explain why C(n, n − 1) = C(n, 1).
19. Quality control wants to test 25 microprocessor chips from the 300 manufactured each day. How many
different batches of test chips are possible?
20. A soccer team carries 18 players on the roster; 11 players make a team. How many different teams are
possible?
21. How many juries of 5 men and 7 women can be formed from a panel of 17 men and 23 women?
22. How many different sets of 4 novels and 3 plays can be created from a collection of 21 novels and
11 plays?
Exercises 23–26 deal with the following situation: Of a company’s personnel, 7 people work in design, 14 in
manufacturing, 4 in testing, 5 in sales, 2 in accounting, and 3 in marketing. A committee of 6 people is to be
formed to meet with upper management.
23. How many committees with 1 member from each department are possible?
24. How many committees with exactly 2 members from manufacturing are possible?
25. How many committees with no representative from accounting and exactly 1 representative from marketing are possible?
26. How many committees with at least 2 representatives from manufacturing are possible?
Exercises 27–32 concern a 5-card hand from a standard 52-card deck. A standard deck has 13 cards from each
of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through10, jack, queen, king, or ace.
Each face value is a “kind” of card. The jack, queen, and king are “face cards.”
27. How many hands contain 4 queens?
28. How many hands contain all diamonds?
29. How many hands contain 3 spades and 2 hearts?
30. How many hands contain cards from all 4 suits?
31. How many hands consist of all face cards?
32. How many hands contain exactly 2 spades and exactly 2 hearts?
Exercises 33–42 concern 5-card poker hands from a standard 52-card deck.
33. How many hands contain a royal straight flush (that is, the 10, jack, queen, king, ace of one suit)?
34. How many hands contain a straight flush (that is, 5 consecutive cards of the same suit, where aces can be
low or high) that is not a royal straight flush (see Exercise 33)?
35. How many hands contain four of a kind (such as 4 jacks plus a fifth card)?
36. How many hands contain a full house (that is, three of a kind plus a pair of another kind)?
37. How many hands contain a flush (that is, 5 cards of the same suit) that is not a straight flush or a royal
straight flush (see Exercises 33 and 34)?
38. How many hands contain a straight (that is, 5 consecutive cards, where aces can be low or high) that is not
a straight flush or a royal straight flush (see Exercises 33 and 34)?
39. How many hands contain three of a kind (that is, exactly 3 cards of the same kind plus 2 other cards that
are not a pair)?
40. How many hands contain 2 pairs (that is, 2 pairs of 2 different kinds plus a fifth card of some third kind)?
41. How many hands contain 1 pair (that is, exactly 2 cards of the same kind)?