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Four women and three men must be seated in a row for a group
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06 Jun 2017, 13:05
06 Jun 2017, 13:05
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Question Stats:
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Four women and three men must be seated in a row for a group photograph. If no two men can sit next to each other, in how many different ways can the seven people be seated?
A) 240
B) 480
C) 720
D) 1440
E) 5640
A) 240
B) 480
C) 720
D) 1440
E) 5640
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Four women and three men must be seated in a row for a group
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07 Jun 2017, 15:02
07 Jun 2017, 15:02
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GreenlightTestPrep wrote:
Four women and three men must be seated in a row for a group photograph. If no two men can sit next to each other, in how many different ways can the seven people be seated?
A) 240
B) 480
C) 720
D) 1440
E) 5640
A) 240
B) 480
C) 720
D) 1440
E) 5640
Take the task of arranging the 7 peopl and break it into stages.
Stage 1: Arrange the 4 women in a row
We can arrange n unique objects in n! ways.
So, we can arrange the 4 women in 4! ways (= 24 ways)
So, we can complete stage 1 in 24 ways
IMPORTANT: For each arrangement of 4 women, there are 5 spaces where the 3 men can be placed.
If we let W represent each woman, we can add the spaces as follows: _W_W_W_W_
So, if we place the men in 3 of the available spaces, we can ENSURE that two men are never seated together.
Let's let A, B and C represent the 3 men.
Stage 2: Place man A in an available space.
There are 5 spaces, so we can complete stage 2 in 5 ways.
Stage 3: Place man B in an available space.
There are 4 spaces remaining, so we can complete stage 3 in 4 ways.
Stage 4: Place man C in an available space.
There are 3 spaces remaining, so we can complete stage 4 in 3 ways.
By the Fundamental Counting Principle (FCP), we can complete the 4 stages (and thus seat all 7 people) in (24)(5)(4)(3) ways (= 1440 ways)
Answer:
Show: ::
D
Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE So be sure to learn this technique.
RELATED VIDEOS
Re: Four women and three men must be seated in a row for a group
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22 Aug 2018, 00:49
22 Aug 2018, 00:49
May I know why is it when we place the men, we don't follow the reasoning where since there are three men, there are three men to choose from, hence 3x, followed by 2x and 1x. Instead, we follow the reasoning of the available chairs instead i.e. 5 chairs, then 4, then 3. I agree with your way as it seems correct but I can't grasp the logic behind it. Thanks for helping me to better understand.
