Question

A boat's crew consists of $8$ men, $3$ of whom can only row on one side and $2$ only on the other. Let the number of ways in which the crew can be arranged be $K^3$ .Find K ?

Answer 1

Let the men $P,Q,R,S,T,U,V,W$ and suppose $P,Q,R$ remain only on one side and $S,T$ on the other as represented in the first figure.
Then, since $4$ men must row on each side, of the remaining $3$, one must be placed on the side of $P,Q,R$ and the other two on the side of $S,T$; this can evidently be done in $3$ ways, for we can place any one of the three on the side of $P,Q,R$.
Now $3$ ways of distributing the crew let us first consider one, way, say that in which $U$ is on the side of $P,Q,R$ as shown in the second figure.
Now $P,Q,R,U$ can be arranged in $4$ ways and $S,T,V,W$ can be arranged in $4!$ ways. Hence total no. pf ways arranging the men
$=4!\times 4!=576$
Hence the number of ways of arranging the crew
$=3\times 576=1728$

$\therefore K=12$