Major Competitive Exams

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Q. How many ways can 4 different prizes be distributed among 3 students?

A. 81
B. 64
C. 27
D. 12

Solution

Each prize can go to any of the 3 students, so the total ways = 3^4 = 81.

Correct Answer: A — 81

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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

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 men and 3 women be arranged in a line if the men must be together? (2019)

A. 5040
B. 720
C. 840
D. 1200

Solution

Treat the 4 men as one unit. Thus, we have 4 units (1 man unit + 3 women) to arrange: 4! * 4! = 24 * 24 = 576.

Correct Answer: A — 5040

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Q. How many ways can 4 people be seated in a row of 6 chairs? (2021)

A. 360
B. 720
C. 240
D. 480

Solution

The number of ways is 6P4 = 6! / (6-4)! = 360.

Correct Answer: A — 360

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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

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 4 students be selected from a class of 10? (2020)

A. 210
B. 120
C. 240
D. 300

Solution

The number of ways to choose 4 students from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. How many ways can 4 students be selected from a group of 10?

A. 210
B. 120
C. 150
D. 180

Solution

The number of ways is C(10, 4) = 210.

Correct Answer: A — 210

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Q. How many ways can 4 students be selected from a group of 12? (2014)

A. 495
B. 144
C. 120
D. 220

Solution

The number of ways to select 4 students from 12 is given by 12C4 = 495.

Correct Answer: A — 495

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Q. How many ways can 4 students be selected from a group of 15?

A. 1365
B. 455
C. 105
D. 210

Solution

The number of ways to select 4 students from 15 is given by 15C4 = 1365.

Correct Answer: A — 1365

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Q. How many ways can 5 different books be arranged on a shelf?

A. 60
B. 120
C. 100
D. 80

Solution

The number of arrangements of 5 different books is 5! = 120.

Correct Answer: B — 120

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Q. How many ways can 5 different books be selected from a shelf of 10 books?

A. 252
B. 120
C. 200
D. 300

Solution

The number of ways is C(10, 5) = 252.

Correct Answer: A — 252

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Q. How many ways can 5 different books be selected from a shelf of 10?

A. 252
B. 120
C. 300
D. 200

Solution

The number of ways to choose 5 from 10 is C(10,5) = 252.

Correct Answer: A — 252

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Q. How many ways can 5 different cards be chosen from a deck of 52 cards? (2019)

A. 2598960
B. 100
C. 1000
D. 500

Solution

The number of ways to choose 5 cards from 52 is given by 52C5 = 2598960.

Correct Answer: A — 2598960

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Q. How many ways can 5 different colored balls be arranged in a line? (2016)

A. 120
B. 60
C. 30
D. 24

Solution

The number of arrangements of 5 different colored balls 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 if each box can hold any number of balls?

A. 243
B. 125
C. 256
D. 3125

Solution

Each ball has 3 choices (boxes), so for 5 balls, the total arrangements = 3^5 = 243.

Correct Answer: A — 243

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Q. How many ways can 5 different flags be arranged on a pole? (2019)

A. 60
B. 120
C. 100
D. 24

Solution

The number of arrangements of 5 different flags is 5! = 120.

Correct Answer: B — 120

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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

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 are identical? (2017)

A. 60
B. 120
C. 30
D. 20

Solution

The number of arrangements of 5 letters where 2 are identical is 5! / 2! = 60.

Correct Answer: B — 120

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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

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 such that two specific letters are always together?

A. 48
B. 60
C. 120
D. 24

Solution

Treat the two specific letters as one unit. Then, we have 4 units to arrange: 4! = 24. The two letters can be arranged in 2! = 2 ways. Total = 24 * 2 = 48.

Correct Answer: B — 60

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Q. How many ways can 5 different letters be arranged such that two specific letters are never together?

A. 48
B. 60
C. 72
D. 96

Solution

Total arrangements = 5! = 120. Arrangements with the two letters together = 4! * 2! = 48. So, arrangements where they are not together = 120 - 48 = 72.

Correct Answer: C — 72

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Q. How many ways can 5 different letters be selected from the alphabet?

A. 26
B. 3003
C. 156
D. 120

Solution

The number of ways to choose 5 letters from 26 is C(26, 5) = 65780.

Correct Answer: B — 3003

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Q. How many ways can 5 different prizes be awarded to 3 students?

A. 60
B. 100
C. 150
D. 200

Solution

The number of ways to award 5 different prizes to 3 students is 3^5 = 243.

Correct Answer: C — 150

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Q. How many ways can 5 people be seated in a row of 5 chairs?

A. 120
B. 60
C. 30
D. 24

Solution

The number of arrangements of 5 people in 5 chairs is 5! = 120.

Correct Answer: A — 120

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Q. How many ways can 5 students be seated in a row of 5 chairs?

A. 120
B. 60
C. 30
D. 90

Solution

The number of ways to arrange 5 students in 5 chairs is 5! = 120.

Correct Answer: A — 120

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Q. How many ways can 5 students be seated in a row of 8 chairs? (2016)

A. 6720
B. 120
C. 240
D. 360

Solution

The number of ways to seat 5 students in 8 chairs is 8P5 = 6720.

Correct Answer: A — 6720

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Q. How many ways can 6 different books be arranged on a shelf if 2 specific books must be together?

A. 120
B. 720
C. 240
D. 480

Solution

Treat the 2 specific books as one unit. Then we have 5 units to arrange: 5! * 2! = 240.

Correct Answer: C — 240

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Q. How many ways can 6 different books be arranged on a shelf? (2017)

A. 720
B. 600
C. 840
D. 960

Solution

The number of arrangements of 6 different books is 6! = 720.

Correct Answer: A — 720

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Q. How many ways can 6 different flags be arranged on a pole? (2023)

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 different flags is 6! = 720.

Correct Answer: A — 720

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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

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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Major Competitive Exams

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