Combinatorics

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.