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Four women and three men must be seated in a row for a group

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Four women and three men must be seated in a row for a group [#permalink] New post 06 Jun 2017, 13:05
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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
[Reveal] Spoiler: OA

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GRE Instructor
User avatar
Joined: 10 Apr 2015
Posts: 582
Followers: 27

Kudos [?]: 376 [0], given: 1

Re: Four women and three men must be seated in a row for a group [#permalink] New post 07 Jun 2017, 15:02
Expert's post
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


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:
[Reveal] Spoiler:
D


Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So be sure to learn this technique.

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Re: Four women and three men must be seated in a row for a group   [#permalink] 07 Jun 2017, 15:02
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