In how many ways can 4 cards be removed from a full deck so that the cards are exactly 3 different suites?

Answer 1

There are 13 cards to a suit, and there are 4 suits. We want exactly one card of one suit, exactly one card of a second suit, and exactly two cards of a third suit.

There are 4C3 ways to choose these three suits.
There are 13C1 ways to chose one card from suit one.
There are 13C1 ways to chose one card from suit two.
There are 13C2 ways to chose two cards from suit three.

Answer: (4C3)(13C1)(13C1)(13C2)

Answer 2

There are four suites Hearts, Spades, Club, Diamonds.
You should select 1 suite to select 2 cards from:

The number of ways to do this is 4*[13!/2!11!] = 4*78 = 312

Now, you should choose 2 suites out of the remaining three from which you select one card, and the number of possibilities to do this is [3!/(2!*1!)] * 13^2 = 3*169 = 507

Now, multiply the two numbers: 312 * 507 = 158,184