Q. How many ways can 4 students be selected from a group of 10?
A.
210
B.
120
C.
150
D.
180
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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 15?
A.
1365
B.
455
C.
105
D.
210
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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
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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
Show solution
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 colored balls be arranged in a line? (2016)
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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
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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 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 are identical? (2017)
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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
Show solution
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 never together?
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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 arranged such that two specific letters are always together?
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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 selected from the alphabet?
A.
26
B.
3003
C.
156
D.
120
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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
Show solution
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?
Show solution
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?
Show solution
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
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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
Show solution
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
Show solution
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
Show solution
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
Show solution
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 6 people be arranged in a circle?
A.
720
B.
120
C.
60
D.
30
Show solution
Solution
The number of arrangements in a circle is (n-1)! = (6-1)! = 5! = 120.
Correct Answer: A — 720
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Q. How many ways can 6 people be divided into 2 groups of 3?
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Solution
The number of ways to divide 6 people into 2 groups of 3 is (6! / (3!3!)) / 2 = 20.
Correct Answer: A — 20
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Q. How many ways can 6 people be seated in a row? (2017)
A.
720
B.
600
C.
480
D.
360
Show solution
Solution
The number of arrangements of 6 people in a row is 6! = 720.
Correct Answer: A — 720
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Q. How many ways can 6 people be selected from a group of 10? (2020)
A.
210
B.
120
C.
300
D.
150
Show solution
Solution
The number of ways to select 6 from 10 is C(10, 6) = 210.
Correct Answer: A — 210
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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
Show solution
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
Show solution
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
Show solution
Solution
The number of arrangements is 8! = 40320.
Correct Answer: A — 40320
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Q. How many ways can a committee of 3 be formed from 5 people?
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Solution
The number of ways to choose 3 from 5 is C(5,3) = 10.
Correct Answer: A — 10
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Q. How many ways can a committee of 3 be formed from 7 people? (2021)
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Solution
The number of ways to form a committee of 3 from 7 is given by 7C3 = 35.
Correct Answer: B — 35
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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
Show solution
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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