Discussion

What are the chances that no two boys are sitting together for a photograph if there are 5 girls and 2 boys?

Answer: D

Seat the five girls first, then place the boys in the gaps between them so that no two boys are adjacent.

\[ \text{total arrangements} = 7! = 5040 \]

Five girls create six gaps (including the ends). Choose two gaps and seat the boys in them:

\[ 5! \times {}^{6}C_{2} \times 2! = 120 \times 15 \times 2 = 3600 \]

\[ P = \frac{3600}{5040} = \frac{5}{7} \]

The gap method is the standard way to enforce a "no two together" condition.

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