First Ace at Card 14

The first 13 cards must be non-aces and card 14 must be an ace.

Think of it via order statistics: choose positions of the 4 aces among 52 cards.

1. 1. Reframe with combinations

   The event happens if and only if exactly one of the 4 aces is in position 14, and the other 3 aces along with the choice of positions of all 4 aces among the first 13+the rest matter; equivalently, condition on positions of the 4 aces among 52 slots.

2. 2. Direct combinatorial count

   The first ace is at position 14 if and only if positions 1–13 contain no ace (all from the 48 non-aces) and position 14 is an ace. Equivalently choose the 3 remaining aces to be among the remaining $52-14=38$ cards.

   $$P=\frac{\left(\genfrac{}{}{0ex}{}{38}{3}\right)}{\left(\genfrac{}{}{0ex}{}{52}{4}\right)}$$

3. 3. Interpretation

   This matches choosing which 4 of 52 positions hold aces, conditioned so that the smallest is position 14: the other 3 ace positions are chosen from the 38 positions after 14, and positions 1-13 are automatically non-ace.

Answer: C(38,3) / C(52,4)