# Usage of Addition principle or Combination.

The question is, " How many ways are there to draw a heart or a club from an ordinary deck of card ? "

The correct method is to use Addition Principle to solve it .

But can I use Combination to solve it as well ? Technically, it is selecting 13 cards out of the 52 cards without any order.

I tried 52C13 + 52C13 .. the results is not logical at all.

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What do you mean by "ways"? Either I choose the right suit or I don't. 26 of the 52 cards are those I want. – Dario Aug 3 '12 at 13:42
I think the OP meant to ask, simply, what's the probability of drawing a club or a heart out of a deck with 52 cards, otherwise it's not easy to understand what (s)he actually meant. – DonAntonio Aug 3 '12 at 13:46
I am currently on the topic of techniques of counting, which includes Addition principle and combination etc. – TheKraven Aug 3 '12 at 13:50

I tried 52C13 + 52C13 .. the results is not logical at all.

The reason $\binom{52}{13}$ is not logical is because what you're saying is, from the entire deck of cards--i.e. from all 52 cards--be it a heart or a club or a spade or a diamond, pick any combination of 13 cards.

So, one such possibility out of the $\binom{52}{13}$ = 635,013,559,600 ways is the entire spade suit, which does not come near what your problem asks.

There is a way to find it using the choose function. Find out how many cards fit the criteria (in this case 13 hearts and 13 diamonds) and specify how many cards (among these) you want to choose (1 card). So that's $\binom{13 + 13}{1} = \binom{26}{1} = 26$.

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Thanks alot. Very clear explanation. – TheKraven Aug 3 '12 at 16:14

Well of course 52c13 would not work. You are not picking 13 cards out of a pack of 52. You are only picking 1 card out of the 52.

If you did something like 13C1 + 13C1 I think that should work, but I guess that is pretty much addition principle (13+13).

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Hmm this cleared some of confusion. I guess this qn isn't possible to solve using Combination. – TheKraven Aug 3 '12 at 13:53
Yeah sorry, the 13C1 comment is actually kind of an irrelevant statement logically, so you may just have to go with addition principle. – mathguy Aug 3 '12 at 13:55

How many ways are there to draw a heart or a club from an ordinary deck of card ?

Event 1

How many ways to pick a club? $\binom{13}{1}=13$

Event 2

How many ways to pick a heart? $\binom{13}{1}=13$

Event 1 and 2 are mutually exclusive. Hence, the total number of ways of picking a club or a heart is $13+13 = 26$ (addition principal was used)

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