Question

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these four cards are of the same suit.

Answer 1

Step 1. Find the number of ways of choosing $4$ cards from a pack of $52$ playing cards.

Apply the formula: ${}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\left(n-r\right)\mathbf{!}}$

Substitute $r=4\mathrm{and}n=52$ into the formula.

$\begin{array}{rcl}{}^{52}C_4& =& \frac{52!}{4!\left(52-4\right)!}\\ & \Rightarrow & {}^{52}C_4=\frac{52!}{4!\left(48\right)!}\\ & \Rightarrow & {}^{52}C_4=\frac{52·51·50·49}{4·3·2·1}\\ & \Rightarrow & {}^{52}C_4=27025\end{array}$

Therefore, the number of ways of choosing $4$ cards from a pack of $52$ playing cards is $27025$.

Step 2. Find the number of ways of choosing $4$ cards of the same suit.

Since there are $13$ cards for each of $4$ the suit, apply the formula to find the number of ways of choosing $4$ cards of the same suit.

$\begin{array}{rcl}{}^{13}C_4& =& \frac{13!}{4!\left(13-4\right)!}\\ & \Rightarrow & {}^{13}C_4=\frac{13!}{4!\left(9\right)!}\\ & \Rightarrow & {}^{13}C_4=\frac{13·12·11·10}{4·3·2·1}\\ & \Rightarrow & {}^{13}C_4=715\end{array}$

Thus, the number of ways of choosing $4$ cards of the same suit is $715$.

Since there are $4$ suits, the number of ways of choosing $4$ cards for one suit is $4\times 715=2860$.

Therefore, the number of ways of choosing $4$ cards of one suit is $2860$.

Therefore, the number of ways of choosing $4$ cards from a pack of $52$ playing cards is $27025$ and the number of ways of choosing $4$ cards of the one suit is $2860$.