A standard deck of cards consists of four suits, with each suit containing 13 cards, for a total of 52 in all

Question

How many 7-card hands will consist of exactly 4 hearts and 2 clubs 

Answer 1

Hi Angela,

This is a classic combination problem.

We are choosing 7 cards from the deck and we want to know how many ways can we have 4 hearts and 2 clubs. Notice that there is a 7th card that must not be a heart or a club, so it must be a diamond or a spade.

First we calculate the number of ways we can choose 4 hearts, from 13 total hearts. This is ₁₃C₄ = 715 ways.

Then we calculate the number of ways we can choose 2 clubs, from 13 total clubs. This is ₁₃C₃ = 286

Then we calculate the number of ways to choose the final card. There are 26 cards that are not a heart nor a club, so there must be 26 ways to choose 1 of them. :)

So, total there are (715)(286)(26) = 5,316,740 ways. It seems like that number might be too big, right? However, if you calculate the number of ways we can draw 7 cards from a deck,

this is ₅₂C₇ = 133,784,560 ways

Hope this helps,

Jim