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

A) \(\frac{1}{4}\)

B) \(\frac{3}{7}\)

C) \(\frac{1}{2}\)

D) \(\frac{3}{4}\)

E) \(\frac{6}{7}\)
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Prashant Ponde

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Let’s first determine the number of ensembles that include Allison. Suppose that Allison has been chosen for one of the ensembles. The three remaining people in the ensemble can be chosen in 7C3 = (7 x 6 x 5)/(3 x 2) = 35 different ways.

Now, let’s determine the number of ensembles that include both Allison and Barbara. Suppose that they are both chosen as members in one of the ensembles. The remaining two people can be chosen in 6C2 = (6 x 5)/(2 x 1) = 15 different ways.

Thus, 15/35 = 3/7 of all the possible ensembles that include Allison also include Barbara.

Answer: B
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