Counting Question: At Least 12 Hearts in 13-Card Hand

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The discussion focuses on calculating the number of 13-card hands that contain at least 12 hearts. Two cases are considered: one where all 13 cards are hearts, which has only one possibility, and another where there are 12 hearts and one non-heart card. The confusion arises around the choices for the non-heart card, clarified to be 39 options rather than 40. The total number of valid hands is calculated as the sum of 13 ways to choose 12 hearts and 39 ways to choose one non-heart, plus the single way to choose all hearts. This method effectively resolves the counting question.
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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.
 
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lionely said:

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?
 
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
 
lionely said:
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?
 
Sorry aren't there 39 choices for the nonheart? and the number of ways of picking 39 nonhearts would it just be 39 ways?
 
lionely said:
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?
 
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?
 
lionely said:
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!
 
Thanks so much, every time I see these counting questions my brain gets jumbled.
 

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