Question

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Answer 1

Here, we have total cards = 52. We have to select 5 cards containing 1 king and 4 other cards i.e. we have to select 1 king out of 4 kings and 4 other cards out of 48 other cards.

$\therefore$ Number of selections = ${}^4{C}_1{\times }^{48}{C}_4=\frac{4!}{3!1!}\times \frac{48!}{4!44!}$

= $\frac{4\times 3!}{3\times 1}\times \frac{48\times 47\times 46\times 45\times 44!}{4\times 3\times 2\times 1\times 44!}$

= $4\times 2\times 47\times 46\times 45=778320$