Binomial Coefficient Probability Question

Question

completely stuck in this probability question. I know to use Hypergeometric probability but im not sure about what numbers i should be using. Any help would be great. A regular deck of 52 playing cards has 13 ranks in 4 suits. Jack, Queen and King of each suit are face cards. Suppose you are randomly dealt seven cards. What is the probability of getting

(b) Three face cards in the same suit and any four cards in another suit (but all four in the same suit)?

(c) Three face cards not all in the same suit, and any four non-face cards?

Answer 1

Let's start with a hint for (b) and then pause awhile

(Choose suit for face cards)(choose face cards)(choose another suit)(choose 4 cards from it)/ ${52\choose 7}$

$${{4\choose 1}{3\choose3}{3\choose1}{13\choose4}\over{52\choose 7}}$$

Of course, this could have been simplified, e.g. we could have simply counted 4 ways to select all 3 face cards from a suit, but I deliberately gave it step by step.

With the hint given on (c) by drhab, you should be able to do (c) now

Answer 2

Hint on c)

There are $\binom{12}3$ possibilities to choose $3$ face cards. There are $4$ possibilities to choose $3$ face cards all in the same suit.

So there are $\binom{12}3-4$ possibilities to choose $3$ face cards not all in the same suit.