# Five cards are drawn at random from an ordinary deck of 52 cards.?

In how many ways is it possible to draw two red aces and two black cards?

I know the answer is 48 but why?

### 2 Answers

- J. J..Lv 76 years agoBest Answer
The answer is most definitely NOT 48 as you have stated unless you have not provided the correct details.

to get two red aces and two black cards out of 5 cards means that the 5th card must be red.

There are 2C2 or just ONE way to get 2 red aces (ie there are only 2 red aces in a deck)

There are 26C2 ways to get any two black cards = 26 * 25 / 2 = 325 ways

ie the first black card can be any of 26 black cards in the deck

the second black card can be any of the 25 remaining (it cannot be the one just drawn)

Then divide by 2 as eg Black ace of clubs followed by black 3 of clubs is the same pair of cards as the black 3 of clubs followed by the black ace of clubs. It is the same for all other combinations of 2 black cards.

26 * 25 /2 = 325

Therefore there are 1 * 325 ways to get 2 red aces and 2 black cards but these then have to be joined by another red card as 5 cards are required in the question.

Having already drawn 2 red aces there remain 24 other red cards to get one red card from = 24 alternatives

Therefore the total number of ways to get both red aces, 2 black cards and another red

= 1 * 325 * 24 = 7800

how can it possibly be as low as 48?

- PastaBellaLv 76 years ago
Because, in a deck of cards, there are 24 black cards. So, if you add in two specific red cards (aces), then you have double the chance.