Question

How many ways can you choose the 5 cards out of a 52 card deck?

Answer 1

If you take all the possible 5 card draws, and multiply that by the probability all 5 cards are hearts, the result is all the possible 5 heart draws.

The number of ways to draw 5 cards from 52 is a combination...52C5. Remember the formula for combinations:

nCk = n! / (k! * (n-k)!. In this case n=52 and k=5:

52!/(5!*(52-5)!)= 52!/(5!*47!) = (52*51*50*49*48)/(5*4*3*2*1) = 2,598,960.

That's the total number of five-card combinations. But what is the probability that all five cards are hearts? There are 13 hearts in the deck of 52 cards.

P card 1 being a heart is 13/52.

P card 2 being a heart (there are one less card and one less heart) is 12/51

P card 3: 11/50

P card 4: 10/49

P card 5: 9/48

The probability of all five events is the product of their probabilities:

13/52 * 12/51 * 11/50 * 10/49 * 9/48 = 0.000495198079231693.

The number of combinations of five cards, times the probability of all five hearts, gives us the answer we want.

0.000495198079231693*2,598,960=1,287.