Question

Three men have 4 coats, 5 waistcoats and 6 caps. In how many ways can they wear them?

Answer 1

$4$ coats

$5$ waistcoats

$6$ caps

Number of ways in which $3$ Men can wear $4$ coats =

$\frac{4!}{(3-4)!}$ $=\frac{4!}{1!}$ $=\frac{1\times 2\times 3\times 4}{1}=24$ ways

Number of ways in which $3$ Men can wear $5$ waistcoats =

$\frac{5!}{(5-3)!}$ $=\frac{5!}{2!}$ $=\frac{1\times 2\times 3\times 4\times 5}{1\times 2}$ $=3\times 4\times 5=60$ ways

Number of ways in which $3$ Men can wear $6$ caps

$\frac{6!}{(6-3)!}$ $=\frac{6!}{3!}$ $=\frac{1\times 2\times 3\times 4\times 5\times 6}{1\times 2\times 3}$ $=4\times 5\times 6=120$ ways

Total number of ways $=24$ $\times$ $60$ $\times$ $120$ $=172800$