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

$\begin{array}{1 1}(A)\;778320\\(B)\;768320\\(C)\;758220\\(D)\;775320\end{array}$

• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
One ace will be selected from four aces and four cards will be selected from $(52-4)=48$ cards
If $P$ is the required number of ways then,
$P=C(4,1)\times C(48,4)$
$\;\;\;=\large\frac{4!}{1!(4-1)!}\times \frac{4!}{4!(48-4)!}$
$\;\;\;=\large\frac{4(3!)}{1!(3)!}\times \frac{48\times 47\times 46\times 45\times 44!}{4\times 3 \times 2 \times 1\times44!}$
$\;\;\;\;=4\times 2\times 47\times 46\times 45$
$\;\;\;=778320$ ways