Combinatorics - Cards

Question

I am struggling a little to wrap my head around this combo problem. If I took 13 cards from a standard deck of playing cards, what would be the chance of having exactly one 10 within the 13 cards?

Answer 1

$\frac{\binom{4}{1}\binom{48}{12}}{\binom{52}{13}}$

${4 \choose 1}$ is the choosing one card $10$

${48 \choose 12}$ is the choosing $12$ cards different from $10$

Answer 2

Total ways to select $13$ cards out of $52$ cards is $52\choose{13}$ So these are the total number of outcomes.
Now we need favorable outcomes go for this $10$ you want now there are 4 ways to choose to get one $10$ One out of the spades,diamonds,hearts, clubs now for your Other 12 cards you have to choose you are left with $(52 - 4 = 48)$ cards to choose from which is indeed
$48\choose{12}$ Now the final expression for getting Probability when each outcome is equally favourable is $$\frac{Favourable Outcomes}{Total Number of Outcomes}$$

$$\frac{ 4\choose{1}}{52\choose{13}}* {48\choose{12}} $$