Permutations & Combinations
Q. How many different 4-digit PINs can be formed using the digits 0-9 if digits cannot be repeated?
A.
5040
B.
10000
C.
9000
D.
7200
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Solution
The first digit can be any of 10 digits, the second can be any of 9, the third can be any of 8, and the fourth can be any of 7. Total = 10 * 9 * 8 * 7 = 5040.
Correct Answer: A — 5040
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Q. How many different 4-digit PINs can be formed using the digits 0-9 without repetition?
A.
5040
B.
9000
C.
10000
D.
1000
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Solution
The number of 4-digit PINs is P(10, 4) = 5040.
Correct Answer: A — 5040
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Q. How many different ways can 3 boys and 2 girls be seated in a row?
A.
30
B.
60
C.
120
D.
180
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Solution
The number of ways to arrange 5 people is 5! = 120.
Correct Answer: C — 120
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Q. How many different ways can the letters of the word 'MATH' be arranged?
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Solution
The number of arrangements of the letters in 'MATH' is 4! = 24.
Correct Answer: B — 24
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Q. How many ways can 2 boys and 3 girls be selected from 5 boys and 6 girls?
A.
100
B.
150
C.
200
D.
250
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Solution
The number of ways is 5C2 * 6C3 = 10 * 20 = 200.
Correct Answer: B — 150
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Q. How many ways can 2 students be selected from a group of 5? (2019)
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Solution
The number of ways to select 2 students from 5 is given by 5C2 = 10.
Correct Answer: A — 10
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Q. How many ways can 3 books be selected from a shelf of 10 books? (2022)
A.
120
B.
210
C.
100
D.
30
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Solution
The number of ways to select 3 books from 10 is given by 10C3 = 120.
Correct Answer: B — 210
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Q. How many ways can 3 boys and 2 girls be seated in a row?
A.
30
B.
60
C.
120
D.
180
Show solution
Solution
The number of ways to arrange 5 people is 5! = 120.
Correct Answer: C — 120
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Q. How many ways can 3 different trophies be awarded to 5 students? (2022)
A.
60
B.
100
C.
120
D.
30
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Solution
The number of ways to award 3 trophies to 5 students is given by P(5,3) = 5! / (5-3)! = 60.
Correct Answer: C — 120
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Q. How many ways can 3 out of 8 different colored balls be chosen? (2022)
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Solution
The number of ways to choose 3 balls from 8 is given by 8C3 = 56.
Correct Answer: A — 56
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Q. How many ways can 3 prizes be awarded to 10 students if no student can win more than one prize?
A.
720
B.
1000
C.
1200
D.
100
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Solution
The number of ways to award 3 prizes is P(10, 3) = 720.
Correct Answer: A — 720
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Q. How many ways can 3 students be selected from a group of 10?
A.
120
B.
90
C.
100
D.
80
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Solution
The number of ways to select 3 students from 10 is given by 10C3 = 120.
Correct Answer: A — 120
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Q. How many ways can 4 different fruits be selected from 8 available fruits? (2021)
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Solution
The number of ways to choose 4 fruits from 8 is given by 8C4 = 70.
Correct Answer: A — 70
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Q. How many ways can 4 different fruits be selected from 8?
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Solution
The number of ways to select 4 fruits from 8 is given by 8C4 = 70.
Correct Answer: B — 56
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Q. How many ways can 4 letters be chosen from the word 'COMBINATION'?
A.
210
B.
120
C.
150
D.
180
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Solution
The number of ways to choose 4 letters from 11 is 11C4 = 330.
Correct Answer: A — 210
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Q. How many ways can 4 prizes be distributed among 10 students if each student can receive at most one prize? (2019)
A.
5040
B.
720
C.
1000
D.
1200
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Solution
The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.
Correct Answer: A — 5040
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Q. How many ways can 5 different letters be arranged if 2 letters are always together?
A.
48
B.
120
C.
240
D.
720
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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 — 240
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Q. How many ways can 5 different letters be arranged if 2 letters must always be together?
A.
60
B.
120
C.
240
D.
720
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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 — 240
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Q. How many ways can 6 different letters be arranged if 2 letters are always together?
A.
120
B.
240
C.
720
D.
1440
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Solution
Treat the 2 letters as one unit. So, we have 5 units to arrange: 5! = 120. The 2 letters can be arranged in 2! = 2 ways. Total = 120 * 2 = 240.
Correct Answer: D — 1440
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Q. How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?
A.
720
B.
120
C.
5040
D.
840
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Solution
Treat the 3 specific books as one unit. So, we have 5 units to arrange: 5! * 3! = 360.
Correct Answer: C — 5040
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Q. How many ways can 8 different books be arranged on a shelf if 3 specific books must be together?
A.
720
B.
5040
C.
40320
D.
2880
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Solution
Treat the 3 specific books as one unit. So, we have 6 units to arrange: 6! = 720. The 3 books can be arranged in 3! = 6 ways. Total = 720 * 6 = 4320.
Correct Answer: B — 5040
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Q. How many ways can 8 different colored balls be arranged in a line?
A.
40320
B.
720
C.
1000
D.
100
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Solution
The number of arrangements is 8! = 40320.
Correct Answer: A — 40320
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Q. How many ways can a committee of 4 be formed from 10 people? (2019)
A.
210
B.
120
C.
100
D.
30
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Solution
The number of ways to form a committee of 4 from 10 is given by 10C4 = 210.
Correct Answer: A — 210
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Q. In how many ways can 3 prizes be awarded to 5 students? (2021)
A.
60
B.
100
C.
120
D.
80
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Solution
The number of ways to award 3 prizes to 5 students is 5P3 = 60.
Correct Answer: C — 120
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Q. In how many ways can 4 different flags be arranged on a pole? (2015)
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Solution
The number of arrangements of 4 distinct flags is 4! = 24.
Correct Answer: A — 24
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Q. In how many ways can 4 different fruits be selected from 10 fruits?
A.
210
B.
120
C.
240
D.
300
Show solution
Solution
The number of ways to choose 4 fruits from 10 is C(10, 4) = 210.
Correct Answer: A — 210
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Q. In how many ways can 4 different prizes be distributed among 10 students? (2023)
A.
5040
B.
10000
C.
2400
D.
120
Show solution
Solution
The number of ways to distribute 4 different prizes among 10 students is 10P4 = 5040.
Correct Answer: A — 5040
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Q. In how many ways can 4 prizes be distributed among 10 students?
A.
210
B.
5040
C.
10000
D.
1001
Show solution
Solution
The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.
Correct Answer: B — 5040
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Q. In how many ways can 5 different books be arranged on a shelf? (2021)
A.
60
B.
120
C.
240
D.
720
Show solution
Solution
The number of arrangements of 5 distinct books is 5! = 120.
Correct Answer: D — 720
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Q. In how many ways can 5 different colored balls be arranged in a line?
A.
60
B.
120
C.
240
D.
720
Show solution
Solution
The number of ways to arrange 5 different colored balls is 5! = 120.
Correct Answer: D — 720
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