# If I draw 5 cards from a deck, how many ways can one get two cards from the same value card?

If I draw five cards from a deck of 51 cards, how many possible arrangements possible if i want 2 cards with the same value ie 2 jacks?

My method in solving this was to do the following: (52)(3)(50C3)

Because the first card doesn't matter and the next one matters and the rest of the three don't matter at all.

I have a very strong feeling that I was overcounting and my method was flawed

i was wondering if I was making any mistakes?

• Is it exactly two cards with the same value? At least two cards with the same value? Is the value fixed beforehand (say, you're only interested in hands which contain exactly/at least tow jacks), or is any repeated value okay? – Fimpellizieri May 24 '17 at 4:47
• I agree with @Fimpellizieri you have to specify more clearly. – Iti Shree May 24 '17 at 4:49
• Did you hide an ace from that deck? :-P – lesath82 May 24 '17 at 6:28
• Did you mean from a deck of $\color{red}{52}$ cards? – N. F. Taussig May 24 '17 at 8:41

Of the $\binom{52}5$ poker hands, there are $$13\times\binom{12}3\times 6\times 4^3$$ which are one pair hands. This is $13$ possible denominations for the pair, $\binom{12}3$ for the singletons, $6=\binom42$ choices for the suits of the pair, and $4^3$ choices for the suits of the singletons.
The number of hands with at least one repeated value is $$\binom{52}5-\binom{13}5 4^5.$$