Question

In how many ways can a pack of 52 cards be divided into 4 sets, three of them having 16 cards each and the fourth just 4 cards?

• A. $16!\times 52!$
• B. $\frac{(52!)}{(16!{)}^3}$
• C. $\frac{52!}{(3!{)}^{16}}$
• D. $\frac{52!}{(16!{)}^3\times (3!)}$

Answer 1

The correct option is D $\frac{52!}{(16!{)}^3\times (3!)}$

First player can get 16 cards in ${}^{52}{C}_{16}$ ways.
Second player can get 16 cards in ${}^{36}{C}_{16}$ ways.
Third player can get 16 cards in ${}^{20}{C}_{16}$ ways.
and fourth player can get 4 cards in ${}^4{C}_4$ ways.
But since first three can be interchanged in 3! ways.
Hence, the required number of ways.
${}^{52}{C}_{16}{\times }^{36}{C}_{16}{\times }^{20}{C}_{16}{\times }^4{C}_4\times \frac{1}{3!}=\frac{52!}{(16!{)}^3\times (3!)}$