Question

Three cards are drawn at random from a pack of $52$ cards.

What is the chance of drawing three aces?

• A. $\frac{3}{5525}$
• B. $\frac{2}{5525}$
• C. $\frac{1}{5525}$
• D. None of these.

Answer 1

The correct option is C

$\frac{1}{5525}$

Explanation of the correct option:

We know that, to find the number of ways for the arrangement of objects where the order in which the objects are selected does not matter is calculated by using the combination.

${}^n{C}_r$ is calculated as, $\frac{n!}{r!(n-r)!}$

We want to choose $3$ cards from $52$ cards. So the total number of arrangements will be calculated by using the combination.

Total cases $={}^{52}C_3$

$$\begin{gathered} =\frac{52!}{3!49!} \\ =\frac{52\times 51\times 50\times 49!}{3\times 2\times 49!} \\ =22100 \end{gathered}$$

Our favorable case is three aces card. we have $4$ aces card and we want to choose $3$.So the total number of arrangements will be calculated by using the combination.

Favorable cases $={}^{4}C_3$

$$\begin{gathered} =\frac{4!}{3!(4-3)!} \\ =\frac{4\times 3!}{3!} \\ =4 \end{gathered}$$

Since, the probability is given by, $\frac{No.offavorablecase}{Totalno.ofcases}$

Therefore the required probability$=\frac{4}{22100}$

The required probability$=\frac{1}{5525}$

Hence option (C) is the correct option.