Why divide by 4! in the this combination?

Question

Q)In how many ways can a pack of 52 cards be formed into 4 groups of 13 cards each? According to me the answer must be $$ C^{52}_{13} *C^{39}_{13} *C^{26}_{13} *C^{13}_{13} * 4! $$ as the order is not important and there i saw other case of rearranging the 4 groups among themselves. However, the in the answer, $$ \frac{C^{52}_{13} *C^{39}_{13} *C^{26}_{13} *C^{13}_{13}}{ 4!} $$ is given as the answer. Why do we have to divide it rather than multiply it?

Answer 1

Consider you want to group the numbers $1,2,3,4$ into two groups with 2 numbers each. Then the two following configurations are equal:

• One group contains $1,2$ and another group contains $3,4$
• One group contains $3,4$ and another group contains $1,2$

And thus you count them as one, or equivalently, you divide by the number of ways in which you can sort the resulting groups.

Answer 2

Because the order of the groups is not important you must count fewer of the arrangements.