Girls not adjacent in a row

Text

$4$ boys and $3$ girls (all distinct) are arranged in a row in random order. What is the probability that no girl is adjacent to another girl?

Solution

• Possible cases: all arrangements of the $7$ people: $7! = 5040$.
• Favorable cases: first arrange the $4$ boys in a row ($4!$ ways); they create $5$ “gaps” ($\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_$) in which to place the girls, at most one per gap. The $3$ distinct girls occupy $3$ of the $5$ gaps in order: $5\cdot 4\cdot 3 = 60$ ways. So the favorable cases are $4!\cdot 60 = 24\cdot 60 = 1440$.

Therefore $P = \frac{1440}{5040} = \frac{2}{7} \approx \boxed{0.286}.$