Three men have 4 coats, 5 waistcoats and 6 caps. In how many ways can they wear them?
$4$ Coats can be selected by $3$ men in $\;$ ${^{4}}{P}_{3} = 4! = 24$ ways
$5$ Waistcoats can be selected by $3$ men in
${^{5}}{P}_{3} = \dfrac{5!}{\left(5 - 3\right)!} = \dfrac{5 \times 4 \times 3 \times 2! }{2!} = 60$ ways
$6$ Waistcoats can be selected by $3$ men in
${^{6}}{P}_{3} = \dfrac{6!}{\left(6 - 3\right)!} = \dfrac{6 \times 5 \times 4 \times 3! }{3!} = 120$ ways
$\therefore \;$ Total number of ways of selecting $4$ coats, $5$ waistcoats and $6$ caps by $3$ men
$= 24 \times 60 \times 120 = 172800$ ways