Combinatorics

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abhirup1711: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Wed Jul 13, 2011 2:27 pmFollowed by: 2 members

by abhirup1711 » Fri Jun 07, 2013 4:27 am

Allison and Barbara are part of an 8 member dance troupe.For a recital,the troupe will be divided into two 4 person ensembles. What fraction of all the possible ensembles that include Allison will also include Barbara?

1/4
3/7
1/2
3/4
6/7

Please help

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Source: Beat The GMAT — Problem Solving | Fri Jun 07, 2013 4:27 am

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by Brent@GMATPrepNow » Fri Jun 07, 2013 5:42 am

abhirup1711 wrote:Allison and Barbara are part of an 8 member dance troupe.For a recital,the troupe will be divided into two 4 person ensembles. What fraction of all the possible ensembles that include Allison will also include Barbara?

A) 1/4
B) 3/7
C) 1/2
D) 3/4
E) 6/7

Here's one approach:

Asking,"What fraction of all the possible ensembles that include Allison will also include Barbara?" is the same as asking, "What is the probability that Allison are Barbara are in the same troupe?

Step 1: Place Allison in one of the troupes.

Step 2: Choose the 3 remaining people to be in Allison's troupe.
Ask, "What is the probability that Barbara is one of the 3 chosen?"
There are 7 people who can fill the remaining 3 spots in Allison's troupe.
So, Barbara has a 3/7 chance of being in Allison's troupe.

Answer = B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by GMATGuruNY » Fri Jun 07, 2013 8:56 am

abhirup1711 wrote:Allison and Barbara are part of an 8 member dance troupe.For a recital,the troupe will be divided into two 4 person ensembles. What fraction of all the possible ensembles that include Allison will also include Barbara?

1/4
3/7
1/2
3/4
6/7

Please help

Brent's solution is great. Here's an alternate approach:

There are 7 people besides Allison.
From these 7 people, 3 must be selected to join Allison's ensemble.

P(Barbara is selected) = 1 - P(Barbara is NOT selected).

P(Barbara is not selected):
P(1st person selected is not Barbara) = 6/7. (Of the 7 people besides Allison, 6 are not Barbara.)
P(2nd person selected is not Barbara) = 5/6. (Of the 6 remaining people, 5 are not Barbara.)
P(3rd person selected is not Barbara) = 4/5. (Of the 5 remaining people, 4 are not Barbara.)
Since all of these events must happen in order for Barbara not to be selected, we multiply the fractions:
6/7 * 5/6 * 4/5 = 4/7.

Thus:
P(Barbara is selected) = 1 - 4/7 = 3/7.

The correct answer is B.

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