Question

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

Answer 1

Finding number of ways of selecting $5$ cards, each selection has exactly one king.

Number of king cards $=4$

Number of other cards $=48$

So, Number of ways of choosing a king $={}^4{C}_1$

Number of ways of choosing other cards $={}^{48}{C}_4$

$\therefore$ Total number of ways $={}^4{C}_1\times {}^{48}{C}_4$

$=\frac{4!}{1!(4-1)!}\times \frac{48!}{4!(48-4)!}$

$=\frac{4!}{3!}\times \frac{48!}{4!\times 44!}$

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

$=\frac{48\times 47\times 46\times 45}{3\times 2}$

$=8\times 47\times 46\times 45=778320$

Hence, total number of ways $=778320$