I'm designing a game with a custom deck. The deck has 5 suits (hearts, spades, clubs, diamonds and crosses) and 5 ranks (1 to 5) for 25 cards total. I'm trying to calculate the odds of revealing 3 cards of the same suit if I draw 10 cards total.

In my head, that math would be: $$ \frac{{5 \choose 1}{5 \choose 3}{20 \choose 7}}{{25 \choose 10}} $$

Where ${5 \choose 1}$ is to pick a suit, ${5 \choose 3}$ is to pick any 3 cards from that suit, ${20 \choose 7}$ is to pick any random 7 cards from the remaining 4 suits, and ${25 \choose 10}$ is the total number of ways to pick 10 cards from a deck of 25. The problem is that my numerator ends up being $3,876,000$, and the denominator is $3,268,760$. Obviously I expect the answer to be very close to 100%, but in this case the probability is greater than 1 (which makes no sense, as you could draw 2 cards from each of all 5 suits and not have met the condition).

What am I doing wrong?