Number of ways to choose 6 cards with the same suit from a normal deck of cards

Question

In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present? One way was $\binom{13}{1}\binom{13}{1}\binom{13}{1}\binom{13}{1}$ but it involves repetition of cases. Other way is total - anti-cases which is quite lengthy. Is there any elegant way to solve this?

Answer 1

Two cases must be discerned:

1) from $1$ of the $4$ suits $3$ cards are chosen.

Leading to: $$\binom{4}{1}\times\binom{13}{3}\binom{13}{1}\binom{13}{1}\binom{13}{1}$$ possibilities.

2) from $2$ of the $4$ suits $2$ cards are chosen.

Leading to: $$\binom{4}{2}\times\binom{13}{2}\binom{13}{2}\binom{13}{1}\binom{13}{1}$$ possibilities.