What is the probability that you draw 2 cards and they sum 21?

Question

An ace is always worth 11, all face cards (Jack, Queen, King) are worth 10, and all number cards are worth the number they show. Given a shuffled deck of 52 cards what is the probability that you draw 2 cards and they sum $21$?

I have counted the number of possible hands $(52C2) = 1326$ and then the combination that adds to $21$ which is $10+11$, there are 16 cards of value 10: 4 10s, Js, Qs, Ks, and then 4 cards of value 11: 4 aces.

So the probability that the 2 drawn cards sum to $21$:

$(16*4)/1326$

However, that is an incorrect answer...

Answer 1

You're right. The probability is $$ \frac{64}{1326} $$ However, depending on how this is graded, they might expect an answer like $$ \frac{32}{663},\quad4.8\% $$ or something similar. The other possibility is that whoever marks your answer is just wrong. It happens from time to time. Or, as it turns out in the comments, $16\cdot 4$ can quickly become $66$ if you're being a bit too quick.

Answer 2

Considering ordered hands,

• total hands: $52\cdot51$.

• winning hands: $4\cdot16+16\cdot4.$