Amount of ways I can pick cards in a deck.

Question

Question: Using all of the face cards in a standard deck of playing cards, in how many ways can I make three piles of five cards.

Thanks :) Been trying for a while now and I can't do it..

Answer 1

You have 15 different cards. To make the first pile, you choose 5 of them, and there are ${15 \choose 5} = 3003$ ways to do that.

You now have 10 cards left. To make the second pile, you choose 5 of them, and there are ${10 \choose 5} = 252$ ways to do that.

You now have 5 cards left, and there's only 1 way to choose all of them to make the third pile.

So there are $3003 \times 252 \times 1 = 756756$ ways to pick your three piles.

However, there is a complication - this assumes that it's important which pile is the first pile, which is the second, and which is the third. If you just care about splitting the cards into three piles, then you have to remove the repetition that comes from being able to swap around which pile is which. In fact, each way of dividing the cards has 6 equivalent ways of organising the piles around, so our count of different ways to pick the piles is 6 times too big. Thus, if order of piles doesn't matter, there are $756756 \div 6 = 126126$ ways to do that.