Question

how many different 4 - card hands can be dealt from a deck of 52 cards? the order of the cards does not matter in this case. your answer is :

Explanation:

Step1: Recall combination formula

The formula for combinations is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items, and $r$ is the number of items to be chosen. Here, $n = 52$ (total number of cards) and $r=4$ (number of cards in a hand).

Step2: Calculate factorial values

$n!=52!$, $r!=4!$, and $(n - r)!=(52 - 4)!=48!$. Then $C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52!}{4!×48!}$. Since $n!=n\times(n - 1)\times\cdots\times1$, we can write $\frac{52!}{4!×48!}=\frac{52\times51\times50\times49\times48!}{4\times3\times2\times1\times48!}$.

Step3: Simplify the expression

Cancel out the $48!$ terms. Then $\frac{52\times51\times50\times49}{4\times3\times2\times1}=\frac{52\times51\times50\times49}{24}=270725$.

Answer:

270725