How many 13-card hands contain at least 12 hearts?

[lionely]

Homework Statement

How many different hands of 13 cards can be obtained containing at least 12 hearts?

Homework Equations

The Attempt at a Solution

I'm confused, well I know there could be a possibility of getting 13 hearts, but I am getting confused trying to work out the other possibilities.

 

[Dick]

Homework Statement

How many different hands of 13 cards can be obtained containing at least 12 hearts?

Homework Equations

The Attempt at a Solution

I'm confused, well I know there could be a possibility of getting 13 hearts, but I am getting confused trying to work out the other possibilities.

The other possibility is choosing 12 out of the 13 hearts and then chosing a final card that's not a heart. Why is that hard to count?

 

[lionely]

Well, because wouldn't each of the 40 cards left to be chosen from be a choice? I'm not sure how to express it

 

[Dick]

Well, because wouldn't each of the 40 cards left to be chosen from be a choice? I'm not sure how to express it

Not if you split into two cases. In the first case you choose 13 hearts. There's only one way to do that. In the second case there are exactly 12 hearts and one card that's not a heart. There aren't 40 choices for the nonheart. How many choices are there?

 

[lionely]

Sorry aren't there 39 choices for the nonheart? and the number of ways of picking 39 nonhearts would it just be 39 ways?

 

[Dick]

Sorry aren't there 39 choices for the nonheart? and the number of ways of picking 39 nonhearts would it just be 39 ways?

You aren't picking 39 nonhearts, you are only picking 1. But yes, there are 39 ways to do that. Now how many ways to pick 12 hearts out of 13?

 

[lionely]

Sorry , for my poor English. Well to pick 12 hearts out of 13... that would be 13C12 so... 13 ways.
So the total amount of ways would be (13 x 39) + that one way of picking all 13 hearts?

 

[Dick]

Sorry , for my poor English. Well to pick 12 hearts out of 13... that would be 13C12 so... 13 ways.
So the total amount of ways would be (13 x 39) + that one way of picking all 13 hearts?

Exactly!

 

[lionely]

Thanks so much, every time I see these counting questions my brain gets jumbled.