Martians and Jovians

Question

In how many ways can five distinct Martians and eight distinct Jovians wait in line if no two Martians stand together?

Answer 1

Make a lineup of $8$ letters $J$ like this: $$J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad$$ There are $7$ gaps between $J$'s that we could slip an $M$ into, plus the $2$ "endgaps," for a total of $9$ places.

So places for the Martians can be chosen in $\binom{9}{5}$ ways. Multiply by $8!5!$ because these are distinct individuals. So once we have chosen the places for the Jovians, and the places for the Martians, we can insert the Jovians in $8!$ orders, and for each way we can insert the Martians in $5!$ different orders.

Answer 2

HINT: In the skeleton $_J_J_J_J_J_J_J_J_$, where each J is a Jovian, the $5$ Martians must occupy $5$ different blanks.