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 $^4C_1$ ways and the remaining 4 cards can be selected out of the 48 cards in $^{48}C_4$ ways.
$=\space^{48}C_4\times\space^4C_1=\frac{48!}{4!44!}\times\frac{4!}{1!3!}$

$=\frac{48\times47\times46\times45}{4\times3\times2\times1\times4}$

Thus, by multiplication principle, required number of 5 card combinations=778320