I need an explanation on the following problem:

If you are dealt $3$ cards from a shuffled deck of $52$ cards, find the probability that all $3$ cards are picture cards.

I found the total number of possible combinations, $\binom{52}3=22100$. now my books uses $\binom{12}3$ to find the number of ways to select $3$ picture cards, which is $220$.

My question is where does the $12$ come from? Why $12$ and not $16$ (number of picture cards on the deck of $52$)?

Thank you


  • 6
    $\begingroup$ There are 12 "picture cards", kings, queens, and jacks. An Ace isn't a picture I suppose. $\endgroup$ May 13, 2012 at 18:39
  • $\begingroup$ Aces don't count. $\endgroup$
    – Double AA
    May 13, 2012 at 18:40
  • $\begingroup$ Thank you, it really makes sense now, I was counting the Ace as a picture card. $\endgroup$
    – Jorge Reis
    May 13, 2012 at 18:54
  • 1
    $\begingroup$ Someone voted to close this as off-topic? Sorry, I don't understand the reasoning there. $\endgroup$ May 14, 2012 at 6:09

1 Answer 1


This is a matter of semantics. What is a "picture card"? I surmise the book thinks it's the queens, kings, and jacks (giving twelve); but not the aces (but not sixteen).

On the other hand, I would consider many of these aces to be picture cards. So your interpretation that there are sixteen picture cards might be justified. I suppose it depends on your experience with playing cards. In any case, in my opinion, the problem is slightly vague.

If in doubt of what some particular phrase means in a problem, it would be a good idea to state what your assumptions are in your write up of a solution. Here, if you were adamant in your stance that aces were picture cards, I would say "I take as the picture cards, the queens, kings, jacks, and aces".


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