Permutation and Combination
Q. From a group of 8 people, how many ways can a committee of 3 be formed?
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Solution
The number of ways to form a committee of 3 from 8 is C(8, 3) = 56.
Correct Answer: A — 56
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Q. From a group of 8 people, how many ways can a committee of 4 be formed?
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Solution
The number of ways to form a committee of 4 from 8 is C(8, 4) = 70.
Correct Answer: A — 70
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Q. How many different 4-digit PIN codes can be formed using the digits 0-9 if digits cannot be repeated?
A.
5040
B.
10000
C.
9000
D.
1000
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Solution
The number of different 4-digit PIN codes is P(10, 4) = 10! / (10-4)! = 5040.
Correct Answer: A — 5040
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Q. How many different 4-digit PIN codes can be formed using the digits 0-9 if repetition is allowed?
A.
10000
B.
1000
C.
5000
D.
9000
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Solution
The number of 4-digit PIN codes is 10^4 = 10000.
Correct Answer: A — 10000
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Q. How many different 4-digit PIN codes can be formed using the digits 0-9 without repetition?
A.
5040
B.
10000
C.
9000
D.
1000
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Solution
The number of different 4-digit PIN codes is P(10, 4) = 10! / (10-4)! = 5040.
Correct Answer: A — 5040
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Q. How many different ways can 2 boys and 2 girls be selected from a group of 5 boys and 4 girls?
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Solution
The number of ways is C(5, 2) * C(4, 2) = 10 * 6 = 60.
Correct Answer: A — 60
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Q. How many different ways can 3 letters be chosen from the word 'APPLE'?
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Solution
The number of ways to choose 3 letters from 'APPLE' (considering 'P' is repeated) is C(5, 3) = 10.
Correct Answer: A — 10
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Q. How many different ways can 3 letters be chosen from the word 'COMPUTER'?
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Solution
The number of ways to choose 3 letters from 8 is C(8, 3) = 56.
Correct Answer: A — 56
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Q. How many different ways can 3 students be selected from a group of 10?
A.
120
B.
720
C.
10
D.
100
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Solution
The number of ways to select 3 students from 10 is C(10, 3) = 10! / (3! * (10-3)!) = 120.
Correct Answer: A — 120
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Q. How many different ways can the letters of the word 'BANANA' be arranged?
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Solution
The number of arrangements is 6! / (3!) = 20.
Correct Answer: C — 30
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Q. How many different ways can the letters of the word 'LEVEL' be arranged?
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Solution
The number of arrangements of 'LEVEL' is 5! / (2! * 2!) = 30.
Correct Answer: A — 60
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Q. How many ways can 10 different trophies be awarded to 3 different winners?
A.
1000
B.
720
C.
1200
D.
100
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Solution
The number of ways to award trophies is P(10, 3) = 10! / (10-3)! = 720.
Correct Answer: A — 1000
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Q. How many ways can 2 boys and 3 girls be selected from a group of 5 boys and 6 girls?
A.
100
B.
60
C.
80
D.
120
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Solution
The number of ways is C(5, 2) * C(6, 3) = 10 * 20 = 200.
Correct Answer: A — 100
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Q. How many ways can 3 different prizes be awarded to 5 students?
A.
60
B.
100
C.
120
D.
30
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Solution
The number of ways to award 3 prizes to 5 students is P(5, 3) = 60.
Correct Answer: C — 120
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Q. How many ways can 4 different books be arranged on a shelf if 2 specific books must be together?
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Solution
Treat the 2 specific books as one unit. So, we have 3 units to arrange: (2 books together + 2 other books) = 3! * 2! = 12.
Correct Answer: A — 48
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Q. How many ways can 4 different colored balls be arranged in a row?
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Solution
The number of ways to arrange 4 different colored balls is 4! = 24.
Correct Answer: A — 24
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Q. How many ways can 4 people be seated at a round table?
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Solution
The number of arrangements of 4 people at a round table is (4-1)! = 3! = 6.
Correct Answer: A — 6
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Q. How many ways can 4 people be selected from a group of 12?
A.
495
B.
300
C.
400
D.
600
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Solution
The number of ways to select 4 people from 12 is C(12, 4) = 495.
Correct Answer: A — 495
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Q. How many ways can 5 books be arranged on a shelf?
A.
60
B.
120
C.
100
D.
24
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Solution
The number of arrangements of 5 distinct books is 5! = 120.
Correct Answer: B — 120
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Q. How many ways can 5 different books be chosen from a shelf of 15 books?
A.
3003
B.
5005
C.
1001
D.
1365
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Solution
The number of ways to choose 5 books from 15 is C(15, 5) = 3003.
Correct Answer: A — 3003
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Q. How many ways can 5 different books be selected and arranged on a shelf?
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Solution
The number of ways to select and arrange 5 books is 5! = 120.
Correct Answer: A — 120
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Q. How many ways can 5 different colored balls be placed in 3 different boxes?
A.
243
B.
125
C.
3125
D.
729
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Solution
Each ball can go into any of the 3 boxes, so the total ways = 3^5 = 243.
Correct Answer: A — 243
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Q. How many ways can 5 different letters be arranged if 2 letters must be together?
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Solution
Treat the 2 letters as one unit, so we have 4 units to arrange: 4! * 2! = 48.
Correct Answer: C — 72
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Q. How many ways can 5 different prizes be awarded to 3 students if each student can receive more than one prize?
A.
243
B.
125
C.
81
D.
64
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Solution
Each prize can go to any of the 3 students, so there are 3^5 = 243 ways.
Correct Answer: A — 243
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Q. How many ways can 5 different prizes be distributed among 3 students?
A.
243
B.
125
C.
60
D.
30
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Solution
Each prize can go to any of the 3 students. Therefore, the total ways = 3^5 = 243.
Correct Answer: A — 243
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Q. How many ways can 6 different colored balls be arranged in a row?
A.
720
B.
600
C.
840
D.
480
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Solution
The number of arrangements of 6 different colored balls is 6! = 720.
Correct Answer: A — 720
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Q. How many ways can 6 people be seated in a round table?
A.
720
B.
120
C.
60
D.
30
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Solution
The number of ways to arrange 6 people in a round table is (6-1)! = 5! = 120.
Correct Answer: A — 720
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Q. How many ways can 6 people be seated in a row if two specific people must sit next to each other?
A.
120
B.
720
C.
240
D.
60
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Solution
Treat the two specific people as one unit. Then, we have 5 units to arrange: 5! = 120. The two people can be arranged in 2! ways. Total = 120 * 2 = 240.
Correct Answer: A — 120
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Q. How many ways can 7 different books be arranged on a shelf if 2 specific books must be together?
A.
720
B.
1440
C.
5040
D.
840
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Solution
Treat the 2 specific books as one unit. Then, arrange 6 units: 6! * 2! = 1440.
Correct Answer: B — 1440
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Q. How many ways can a committee of 3 be formed from 8 people?
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Solution
The number of ways to form a committee is C(8,3) = 56.
Correct Answer: A — 56
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