Answer:
24.
Step-by-step explanation:
In a standard deck of 52 cards, there are four kings and four aces.
Permutation [tex]\displaystyle {n \choose k}[/tex] gives the number of ways to choose [tex]k[/tex] items out of a pile of [tex]n[/tex] of these items. The equation for calculating
[tex]\displaystyle {n \choose k} = \frac{n!}{(n - k)! \cdot k!}[/tex].
Consider: how many ways to choose two kings out of four? Assume that the order for choosing the two cards does not matter. The number of possible ways can be found using the following permutation:
[tex]\displaystyle {4 \choose 2} = \frac{4!}{(4 - 2)! \cdot 2!} = \frac{4!}{2! \cdot 2!}=\frac{4 \times 3 \times 2 \times 1}{(2 \times 1) \times (2 \times 1)} = 6[/tex].
Similarly, find the number of ways to choose three aces out of four.
[tex]\displaystyle {4 \choose 3} = \frac{4!}{(4 - 3)! \cdot 3!} = \frac{4!}{1! \cdot 3!}\frac{4 \times 3}{1 \times (3 \times 2 \times 1)} = 4[/tex].
Note that
Therefore, multiply these two permutations to find the number of ways to choose these five cards:
[tex]\displaystyle {4 \choose 2} \cdot {4 \choose 3}= 24[/tex].
