Question

In how many ways can a pack of 52 cards be formed into 4 groups of 13 cards each?

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Here order of group is not important, then the number of ways in which 52 different cards can be divided equally into 4 groups is $\frac{52!}{4!(13!{)}^4}$
$Alternativemethod$: Each group will get 13 cards. Now first group can be given 13 cards out of 52 cards in ${}^{52}{C}_{13}$ ways. Second group can be given 13 cards out of remaining 39 cards (i.e. 52-13=39) in ${}^{39}{C}_{13}$ ways. Third group can be given 13 cards out of remaining 26 cards (i.e., 26-13=13) in ${}^{13}{C}_{13}$ ways. But the all(four) groups can be interchanged in 4! ways. Hence the required number of ways

=${}^{52}{C}_{13}\times {}^{39}{C}_{13}\times {}^{26}{C}_{13}\times {}^{13}{C}_{13}\times \frac{1}{4!}$

=$\frac{52!}{13!39!}\times \frac{39!}{13!26!}\times \frac{26!}{13!13!}\times 1\times \frac{1}{4!}$

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