Question

How many $5$-card poker hands are there?

Short answer

Number of $5$-card poker hands$=25,98,960$.

Step-by-step solution

Step 1. Given information.

No. of cards in a deck $=52=n$

No. of cards to be chosen = $5=r$

This is a counting problem, so we will apply the formula of combination, that is

$\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$

Step 2. Compute the number of 5-card poker hand.

$$\begin{gathered} \left(\begin{array}{c}52\\ 5\end{array}\right)=\frac{52!}{5!47!} \\ =\frac{52\times 51\times 50\times 49\times 48\times 47!}{5\times 4\times 3\times 2\times 1\times 47!} \\ =25,98,960 \end{gathered}$$

Therefore, the number of $5$-card poker hand is$25,98,960$.