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Sets, Combinatorics, and Probability
42. If the joker is added to the deck and functions as a fifth ace (of any suit), how many hands contain a royal
straight flush (that is, the 10, jack, queen, king, ace of one suit)?
For Exercises 43–48, 14 copies of a code module are to be executed in parallel on identical processors organized into two communicating clusters, A and B. Cluster A contains 16 processors and cluster B contains
32 processors.
43. How many different sets of processors can be used?
44. How many different sets of processors can be used if all modules must execute on cluster B?
45. How many different sets of processors can be used if 8 modules are to be processed on cluster A and 6 on
cluster B?
46. How many different sets of processors can be used if cluster A has 3 failed processors and cluster B has
2 failed processors?
47. How many different sets of processors can be used if exactly 2 modules are to execute on cluster B?
48. How many different sets of processors can be used if all modules are to be executed either on cluster A or
on cluster B?
For Exercises 49–52, a set of 4 coins is selected from a box containing 5 dimes and 7 quarters.
49. Find the number of sets of 4 coins.
50. Find the number of sets in which 2 coins are dimes and 2 are quarters.
51. Find the number of sets composed of all dimes or all quarters.
52. Find the number of sets with 3 or more quarters.
Exercises 53–56 concern a computer network with 60 switching nodes.
53. The network is designed to withstand the failure of any 2 nodes. In how many ways can such a failure
occur?
54. In how many ways can 1 or 2 nodes fail?
55. If 1 node has failed, in how many ways can 7 nodes be selected without encountering the failed node?
56. If 2 nodes have failed, in how many ways can 7 nodes be selected to include exactly 1 failed node?
In Exercises 57–60, a congressional committee of 3 is to be chosen from a set of 5 Democrats, 3 Republicans,
and 4 independents.
57. How many committees are possible?
58. How many committees with at least 1 independent are possible?
59. How many committees that do not include both Democrats and Republicans are possible?
60. How many committees with at least 1 Democrat and at least 1 Republican are possible?
In Exercises 61–66, the states of California, Arizona, New Mexico, Utah, and Nevada each send a team of
6 delegates to the Southwestern States annual conference. A subcommittee of 9 is to be formed to discuss water
rights.
61. How many committees are possible?
62. How many committees with no delegates from New Mexico are possible?
63. How many committees with exactly 1 delegate from New Mexico are possible?
64. How many committees with at most 1 delegate from New Mexico are possible?
Sets, Combinatorics, and Probability
42. If the joker is added to the deck and functions as a fifth ace (of any suit), how many hands contain a royal
straight flush (that is, the 10, jack, queen, king, ace of one suit)?
For Exercises 43–48, 14 copies of a code module are to be executed in parallel on identical processors organized into two communicating clusters, A and B. Cluster A contains 16 processors and cluster B contains
32 processors.
43. How many different sets of processors can be used?
44. How many different sets of processors can be used if all modules must execute on cluster B?
45. How many different sets of processors can be used if 8 modules are to be processed on cluster A and 6 on
cluster B?
46. How many different sets of processors can be used if cluster A has 3 failed processors and cluster B has
2 failed processors?
47. How many different sets of processors can be used if exactly 2 modules are to execute on cluster B?
48. How many different sets of processors can be used if all modules are to be executed either on cluster A or
on cluster B?
For Exercises 49–52, a set of 4 coins is selected from a box containing 5 dimes and 7 quarters.
49. Find the number of sets of 4 coins.
50. Find the number of sets in which 2 coins are dimes and 2 are quarters.
51. Find the number of sets composed of all dimes or all quarters.
52. Find the number of sets with 3 or more quarters.
Exercises 53–56 concern a computer network with 60 switching nodes.
53. The network is designed to withstand the failure of any 2 nodes. In how many ways can such a failure
occur?
54. In how many ways can 1 or 2 nodes fail?
55. If 1 node has failed, in how many ways can 7 nodes be selected without encountering the failed node?
56. If 2 nodes have failed, in how many ways can 7 nodes be selected to include exactly 1 failed node?
In Exercises 57–60, a congressional committee of 3 is to be chosen from a set of 5 Democrats, 3 Republicans,
and 4 independents.
57. How many committees are possible?
58. How many committees with at least 1 independent are possible?
59. How many committees that do not include both Democrats and Republicans are possible?
60. How many committees with at least 1 Democrat and at least 1 Republican are possible?
In Exercises 61–66, the states of California, Arizona, New Mexico, Utah, and Nevada each send a team of
6 delegates to the Southwestern States annual conference. A subcommittee of 9 is to be formed to discuss water
rights.
61. How many committees are possible?
62. How many committees with no delegates from New Mexico are possible?
63. How many committees with exactly 1 delegate from New Mexico are possible?
64. How many committees with at most 1 delegate from New Mexico are possible?