Question

The number of ways of dividing $52$ cards amongst four players equally, are?

• A. $\frac{52!}{{\left(13!\right)}^4}$
• B. $\frac{52!}{\left[{\left(13!\right)}^2.4!\right]}$
• C. $\frac{52!}{\left[{\left(12!\right)}^2.4!\right]}$
• D. None of these

Answer 1

The correct option is A

$\frac{52!}{{\left(13!\right)}^4}$

The explanation for the correct answer.

Find the ways of dividing $52$ cards.

Number of ways

$$\begin{gathered} ={}^{52}\mathrm{C}_{13}\times {}^{39}\mathrm{C}_{13}\times {}^{26}\mathrm{C}_{13}\times {}^{13}\mathrm{C}_{13} \\ =\frac{52!}{13!\left(52-13\right)!}\times \frac{39!}{13!\left(39-13\right)!}\times \frac{26!}{13!\left(26-13\right)!}\times \frac{13!}{13!\left(13-13\right)!}\mathbf{}\left({}^n{C}_r=\frac{n!}{r!\left(n-r\right)!}\right) \\ =\frac{52!}{13!\times 39!}\times \frac{39!}{13!\times 26!}\times \frac{26!}{13!\times 13!}\left(\because 0!=1\right) \\ =\frac{52!}{{\left(13!\right)}^4} \end{gathered}$$

Hence option $\left(A\right)$ is the correct answer.