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