Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.
Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.
In a deck of $52$ cards, there are $4$ aces. A combinations of $5$ cards have to be made in which there is exactly one ace.
Then, one ace can be selected in $^{4} C_{3}$ 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 $=\,^{48} C_{4} \times \,^{4} C_{1}=\frac{48 !}{4 ! 44 !} \times \frac{4 !}{1 ! 3 !}$
$=\frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} \times 4 !$
$=778320$
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