Question

Three cards are drawn at random from a pack of $52$ cards. What is the probability that all the three cards are kings?

Answer 1

Total number of cards are $52$.

The number of ways of selecting three cards is:

${}^{52}{C}_3=\frac{52!}{(52-3)!\times 3!}=\frac{52!}{49!\times 3!}=\frac{52\times 51\times 50\times 49!}{49!\times 3\times 2\times 1}=17\times 26\times 50=22100$

Therefore, $n\left(S\right)=22100$

Let $A$ denote the event of drawing $3$ kings, therefore,

$n\left(A\right){=}^4{C}_3=\frac{4!}{(4-3)!\times 3!}=\frac{4!}{1!\times 3!}=\frac{4\times 3\times 2\times 1}{1\times 3\times 2\times 1}=4$

Thus, probability of drawing $3$ kings is:

$P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{4}{22100}=\frac{1}{5525}$

Hence, probability that all the three cards are kings is $\frac{1}{5525}$.