In a game of Skat there are 4 suits (spades, hearts, diamonds, clubs) and 8 values (7, 8, 9, 10, jack, queen, king, ace) yielding 32 cards altogether. I'm trying to figure out in how many ways $k \geq 4$ cards can be picked, such that every suit is represented. This is my approach:
Pick the first 4 cards. For this there are $8^4$ possibilities (8 choices from from each suit).
For the remaining $k-4$ cards to be picked there are $28 \choose k-4$ possibilities.
The order, in which the cards were picked doesn't matter, so divide by $k!$.
In total: $\frac{8^4 \cdot {28 \choose k-4}}{k!}$
Is this correct or am I missing something?