Question

Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.

Answer 1

In a deck of $52$ cards, there are $4$ aces.

A combination of $5$ cards have to be made in which there is exactly one ace.

Then, one ace can be selected in ${}^4{C}_1$ways and the remaining $4$ cards can be selected out of the $48$ cards in ${}^{48}{C}_4$ ways.

Thus, by multiplication principle, required number of $5$ card combinations

$={}^{48}{C}_4{\times }^4{C}_1$
$=\frac{48!}{4!\left(44\right)!}\times \frac{4!}{1!3!}$
$=\frac{48\times 47\times 46\times 45}{4\times 3\times 2\times 1}\times 4$
$=778320$