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

STEP 1 : Finding the number of ways in which $1$ ace and$4$ other cards can be selected

In a deck of $52$ cards, there are $4$ aces.

A combination of $5$ cards has to be made in which there is exactly $1$ ace.

Number of ways in which $1$ ace can be selected out of $4$ aces $={}^{4}C_1$

$\Rightarrow {}^{4}C_1=\frac{4!}{1!\times 3!}=\frac{4\times 3\times 2\times 1}{1\times 3\times 2\times 1}=4$

Number of ways in which remaining $4$ cards can be selected out of $48$ cards $={}^{48}C_4$

$\Rightarrow {}^{48}C_4=\frac{48!}{4!\times 44!}=\frac{48\times 47\times 46\times 45}{4\times 3\times 2\times 1}=\frac{4669920}{24}=194580$

STEP 2 : Finding the number of ways in which $5$ card combinations can be selected

By multiplication principle, the required number of 5 card combinations are

$\Rightarrow {}^{4}C_1\times {}^{48}C_4$

$\Rightarrow 4\times 194580$

$\Rightarrow 778320$

Hence, the number of $5$ card combinations out of a deck of $52$ cards is $778320$.