I'm doing some math problems for fun, and trying out a couple of methods of solving them. I think I am messing up one of them because there is a discrepancy by a factor of 4.
We have a well-shuffled deck of 52 cards, and we want to get a specific suit to appear for the first time on the 4th card. Say spade.
I decided to solve this in a couple of ways.
My first option was to look at each card individually. There are 39 ways to get the first card out of 52 (no spades can appear there). 38 for the second card out of 51. 37 for the third card out of 50. For the fourth card there are 13 ways to get them out of 49. So all of this ends up being equal to $\frac{39*38*37*13}{52*51*50*49}$ which simplifies to $\frac{37*19*13}{49*25*17*4}$.
My second option was to consider the probability of getting 3 non spade cards plus a spade card out of all possible combinations. There are $C_3^{39}$ ways to choose 3 cards out of 39, and $C_1^{13}$ to choose 1 out of 13 possible spades. We consider the probability of getting that out of all possible ways of getting 4 cards out of 52 and we end up with the following equation $\frac{C_3^{39}*C_1^{13}}{C_4^{52}}$ which simplifies to $\frac{37*19*13}{49*25*17}$.
There is a discrepancy by a factor of 4. What am I missing?