Card problem

If 7 cards are drawn from a deck of cards, how many hands have exactly 5 hearts ?

My try at this, though I do not know if I'm right :

H - Heart
NH - Not Heart

One possible combination would be

Now counting the number of possibilities with this pattern : 13*12*11*10*9*39*38
Now multiplying that by the total number of permutations of this particular pattern, I get the final answer
13*12*11*10*9*39*38*7!/(5!2!)

Could you please verify this or correct me if I'm wrong ?
Thank you very much!

WhiteAeons said:

...
Now counting the number of possibilities with this pattern : 13*12*11*10*9*39*38
Now multiplying that by the total number of permutations of this particular pattern, I get the final answer
13*12*11*10*9*39*38*7!/(5!2!) ...

Click to expand...

First of all 13*12*11*10*9*39*38 is a permutation number [order of the elements matter] and what you want to do is reduce this by the number of duplicate sets you could have with this permutation, see
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
Thus for exactly 5 hearts being contained in the 7 drawn from the deck, you would have
\(\displaystyle \frac{\text13*12*11*10*9*39*38}{\text5*4*3*2*1 * 2*1}\) sets of 7 cards containing exactly 5 hearts.
That is, the hearts can be arranged 5! ways and the non-hearts can be arranged 2! ways
Now we come to the point where I am going to assume that you are going to choose 5 cards from the 7 drawn and want to know how many would contain 5 hearts. Well, since we only have 5 hearts, there is only one hand which contains 5 hearts.

How about if we draw exactly 6 hearts out of 7 cards? And how about 7 hearts out of 7 cards? How would you put them together? Putting all of those together, what would be the total of hands with 5 hearts?

Or possibly I didn't understand the question.

Edit to correct formula.

Ishuda said:

First of all 13*12*11*10*9*39*38 is a permutation number [order of the elements matter] and what you want to do is reduce this by the number of duplicate sets you could have with this permutation, see
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
Thus for 5 hearts chosen out of the 7 drawn from the deck, you would have
\(\displaystyle \frac{\text13*12*11*10*9*39*38}{\text7*6*5*4*3*2*1}\)

Click to expand...

I'm sure that is wrong because that isn't an integer. Maybe I must divide my original number by \(\displaystyle {\text5! 2!}\) ?

WhiteAeons said:

If 7 cards are drawn from a deck of cards, how many hands have exactly 5 hearts

Click to expand...

\(\displaystyle \dbinom{13}{5}\cdot\dbinom{39}{2}\)

pka said:

\(\displaystyle \dbinom{13}{5}\cdot\dbinom{39}{2}\)

Click to expand...

That is exactly what I was thinking. Thank you !

WhiteAeons said:

I'm sure that is wrong because that isn't an integer. Maybe I must divide my original number by \(\displaystyle {\text5! 2!}\) ?

Click to expand...

You're correct it is wrong. I corrected the post. There are only 5 hearts so they can be arranged 5! ways, not 7! ways. Oh, and note that the (corrected) answer is equal to ₁₃C₅ * ₃₉C₂

Edited to add: Just exactly what is meant by "If 7 cards are drawn from a deck of cards, how many hands have exactly 5 hearts?" Do you mean that you draw 7 cards from the deck of 52 and then draw 5 (a hand being 5 cards in poker) from them and what is the number having 5 hearts? Or do you mean you have the set of all 7 card hands which can be drawn from a deck of 52 and, out of these how many contain exactly 5 hearts?

Ishuda said:

You're correct it is wrong. I corrected the post. There are only 5 hearts so they can be arranged 5! ways, not 7! ways.

Click to expand...

Thank you for the help