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Bunuel
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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 m 24 Sep 2021, 07:45
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D
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Difficulty:

65% (hard)

Question Stats:

56% (02:03) correct 44% (02:32) wrong based on 213 sessions

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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 must stand in a line from front to back. If at least one of John and Benjamin must be in the line, in how many different ways could the line be created?

A. 96
B. 100
C. 112
D. 120
E. 144
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D
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baruto
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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 m 24 Sep 2021, 08:08
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Bunuel
Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 must stand in a line from front to back. If at least one of John and Benjamin must be in the line, in how many different ways could the line be created?

A. 96
B. 100
C. 112
D. 120
E. 144
Show more

if we chose 4 out of 5 people then at least one of John and Benjamin must be in the line. Hence we can simply ignore this part

Now number of ways to chose 4 people ot of 5 = \(5c4\) = 5
Number of ways of arranging 4 players = 4!
Number of ways line can be created = 5* 4! = 120
Hence D
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QuantMadeEasy
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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 m 25 Sep 2021, 07:15
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Quote:
Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 must stand in a line from front to back. If at least one of John and Benjamin must be in the line, in how many different ways could the line be created?
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Case 1: Ways of choosing one out of John and Benjamin, and choosing remaining three out of three people
2C1 * 3C3 = 2*1 = 2
These four people can rearrange within themselves in 4! ways
Number of ways = 2*4! = 2*24 = 48

Case 2: Ways of choosing both John and Benjamin, and choosing remaining two out of three people
2C2 * 3C2 = 1*3 = 3
These four people can rearrange within themselves in 4! ways
Number of ways = 3*4! = 3*24 = 72

Total number of ways = 48 + 72 = 120

D is correct
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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 m 03 Jan 2025, 10:10
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Bunuel
Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 must stand in a line from front to back. If at least one of John and Benjamin must be in the line, in how many different ways could the line be created?

A. 96
B. 100
C. 112
D. 120
E. 144
Show more
Is the answer just not 5P4 coz since we are choosing 4 of 5 we will definitely have John or Benjamin.
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Bunuel
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Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 m 03 Jan 2025, 10:52
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jsam98
Bunuel
Out of a group of 5 people (Seb, Danny, Benjamin, Mark, and John), 4 must stand in a line from front to back. If at least one of John and Benjamin must be in the line, in how many different ways could the line be created?

A. 96
B. 100
C. 112
D. 120
E. 144
Is the answer just not 5P4 coz since we are choosing 4 of 5 we will definitely have John or Benjamin.
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Yes, the answer is simply 5P4, since we are selecting 4 people out of 5, and in any selection of 4, either John or Benjamin will always be included. Therefore, the constraint in the question is automatically satisfied. This reduces the problem to just choosing 4 out of 5 when order matters, which, as you mentioned, is 5P4 = 120.

Answer: D.
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