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

Question:

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

Solution:

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 ${ }^{4} \mathrm{C}_{1}$ ways and the remaining 4 cards can be selected out of the 48 cards in ${ }^{48} \mathrm{C}_{4}$ ways.

$={ }^{48} \mathrm{C}_{4} \times{ }^{4} \mathrm{C}_{1}=\frac{48 !}{4 ! 44 !} \times \frac{4 !}{1 ! 3 !}$

Thus, by multiplication principle, required number of 5 card combinations  $=\frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} \times 4$

$=778320$