why is the probability of pulling blackjack from a full deck $\frac{4\cdot 16\cdot 2}{52 \cdot 51}$?

Question

I understand that the 4 in the numerator is for the aces and the 16 is for drawing a king, queen, jack or 10. (52*51) in the denominator is all possible permutations of draws. I'm not really sure why the 2 is there? I'm assuming it's because you can draw an ace first then any of the cards that are worth 10 points OR you can draw any of the cards worth 10 points then an ace so that's 2 possible orderings. But I assumed cause of multiplications commutative property that (4 * 16) alone would count for both cases? (Because (4 * 16) = (16 * 4)? Or you have to multiply by 2 because it does not? How does that work?)

Answer 1

The problem is that there aren’t actually $52\cdot 51$ different draws. There are only $\frac{52\cdot 51}{2}=\binom{52}{2}$ different draws, since the order in which you draw your two cards does not matter (getting a king then an ace is the same as getting an ace then a king). Thus, the probability is $$\frac{4\cdot 16}{52\cdot 51/2}=\frac{4\cdot 16\cdot 2}{52\cdot 51}$$ Your intuition is correct.

Answer 2

The numerator should be $$4\cdot 16+16\cdot 4$$ The first counts ace+kqjt, while the second counts kqjt+ace. This can be simplified as $$4\cdot 16\cdot 2$$