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# Ann, Mark, Dave and Paula line up at a ticket window. In how many way

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Intern
Joined: 19 Dec 2009
Posts: 27
Ann, Mark, Dave and Paula line up at a ticket window. In how many way  [#permalink]

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05 Nov 2010, 13:06
00:00

Difficulty:

5% (low)

Question Stats:

83% (00:53) correct 17% (00:37) wrong based on 47 sessions

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Ann, Mark, Dave and Paula line up at a ticket window. In how many ways can they arrange themselves so that Dave is third in line from the window?

a 24
b 12
c 9
d 6
e 3
Manager
Joined: 10 Sep 2010
Posts: 119
Re: Ann, Mark, Dave and Paula line up at a ticket window. In how many way  [#permalink]

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05 Nov 2010, 13:19
D

Dave's position is fixed. Now we need to arrange all others around Dave.
There are 3x2x1= 6 ways to arrange three people.
CEO
Joined: 11 Sep 2015
Posts: 3130
Re: Ann, Mark, Dave and Paula line up at a ticket window. In how many way  [#permalink]

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22 Oct 2018, 08:51
Top Contributor
rtaha2412 wrote:
Ann, Mark, Dave and Paula line up at a ticket window. In how many ways can they arrange themselves so that Dave is third in line from the window?

a 24
b 12
c 9
d 6
e 3

Take the task of arranging the 4 people and break it into stages.

We’ll begin with the most restrictive stage.

Stage 1: Select a place in line for Dave
Since Dave must be in the 3rd position, we can complete stage 1 in 1 way

Stage 2: Select a place in line for Ann
Now that Dave is positioned in line, there are 3 available places remaining.
So we can complete stage 2 in 3 ways

Stage 3: Select a place in line for Mark
Now that Dave and Ann are positioned in line, there are 2 available places remaining.
So we can complete stage 3 in 2 ways

Stage 4: Select a place in line for Paula
Now that Mark, Dave and Ann are positioned in line, there is 1 space remaining.
So we can complete stage 4 in 1 way

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus arrange all 4 people) in (1)(3)(2)(1) ways (= 6 ways)

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

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Joined: 12 Oct 2010
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Re: Ann, Mark, Dave and Paula line up at a ticket window. In how many way  [#permalink]

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22 Oct 2018, 10:34
rtaha2412 wrote:
Ann, Mark, Dave and Paula line up at a ticket window. In how many ways can they arrange themselves so that Dave is third in line from the window?

a 24
b 12
c 9
d 6
e 3

$$?\,\,\,:\,\,\,{\text{# }}\,\,{\text{ways}}\,\,{\text{with}}\,\,{\text{Dave position}}\,\,{\text{fixed}}$$

We may consider Dave does not "exist": put Ann, Mark and Paula in line (at the 3 positions available to them)!

$$? = {P_3} = 3! = 6$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Ann, Mark, Dave and Paula line up at a ticket window. In how many way &nbs [#permalink] 22 Oct 2018, 10:34
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