How many ways are there to pick k cards in a game of Skat?

Question

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?

Answer 1

Use inclusion/exclusion principle:

• Include the number of ways to choose cards from at most $\color\red4$ suits: $\binom{4}{\color\red4}\cdot\binom{8\cdot\color\red4}{k}$
• Exclude the number of ways to choose cards from at most $\color\red3$ suits: $\binom{4}{\color\red3}\cdot\binom{8\cdot\color\red3}{k}$
• Include the number of ways to choose cards from at most $\color\red2$ suits: $\binom{4}{\color\red2}\cdot\binom{8\cdot\color\red2}{k}$
• Exclude the number of ways to choose cards from at most $\color\red1$ suits: $\binom{4}{\color\red1}\cdot\binom{8\cdot\color\red1}{k}$

Hence the total number of ways is:

$$\sum\limits_{n=0}^{3}(-1)^{n}\cdot\binom{4}{4-n}\cdot\binom{8\cdot(4-n)}{k}$$