# How many different ﬂush hands can you have that consist only of hearts, diamonds or spades? [duplicate]

In poker, a ﬂush is a 5-card hand where all the cards have the same suit. Recall that a deck of cards has 52 cards. There are 4 suits (hearts, clubs, diamonds and spades) and each suit has 13 cards.

How many ways can you get a flush of hearts, diamonds or spades?

For this question, I am thinking the answer is 13C5+13C5+13C5 where C is choses.I am not so sure because it looks tricky

## marked as duplicate by Brian M. Scott, Ittay Weiss, Amzoti, robjohn♦Apr 9 '13 at 18:04

• Why the downvote? – Cameron Buie Apr 9 '13 at 16:33
• When asking questions, it is a good idea to put the entire question in the body, rather than split it between the body and the title. – rschwieb Apr 9 '13 at 16:33
• @exploringnet you only want to know all different ways to obtain a flush, not a probability or something, so the no. of players are irrelevant. And you don't know what kind of poker it is. Could be 5 hand, or 2 hand with 5 open card so maybe there isnt a flop.. – Bob Apr 9 '13 at 16:38
• @exploringnet I dont see the problem here. The question is clear. You take 5 cards, what is the number of possibilities to get a flush of diamonds, spades or harts. – Bob Apr 9 '13 at 16:42
• Seems to be the case, I think, yes – Bob Apr 9 '13 at 16:46

It is exactly as you wrote: $13\choose5$ counts the number of different ways to get 5 cards out of that particular suit. This is the same for any suit, and so if you're interested in those particular flushes, the answer is as you gave: $3\cdot {13\choose 5}$.

There are indeed $_{13}C_5$ ways to have a flush hand of any given suit, and flushes of different suits are mutually exclusive events. Thus, just adding them up (as you did) works fine!

That's it =)

So $$3 \cdot {13 \choose 5}$$

• Even this could have been a comment and even have solved the problem to end in comments only. – ABC Apr 9 '13 at 16:55
• It's at least the right answer + confirmation for OP. But I wasn't trying to start a tug-o-war, so get over it. A question needs an answer, people provide answers and don't give a half-relevant answer if they know what correct answer is. That's my view. – Bob Apr 9 '13 at 16:58
• @exploringnet downvoting, now that's mature :') – Bob Apr 9 '13 at 17:05

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥

The total no of flushes possible are:$4\times ^{13}C_5$ and the favourable are $3\times^{13}C_4$
Chance for flush:

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥

http://en.wikipedia.org/wiki/Hand_rankings#Flush

• @Manuel For good discussion Post this on poker.SE – ABC Apr 9 '13 at 16:43
• 1. this is for a flush of any suit, 2. a straight flush is still a flush, 3. its the chance you get a flush, not the number of different flushes – Bob Apr 9 '13 at 16:45
• We both know that, but as this was the question, you maybe should give that as an answer then, or dont give an answer as above at all. You could post it as a comment maybe if you want to make this clear. – Bob Apr 9 '13 at 16:50
• Not only we both but also the OP knows it. and You are no one to ask me to do so.If you feel so,I'd like to say: even you have not got stars down for him in your answer. – ABC Apr 9 '13 at 16:52
• Ok, if you really consider the above answer a relevant... And I'm not asking anything from you, you could see it as a tip. I actually dont mind what you do with it afterwards :) – Bob Apr 9 '13 at 16:55