What is the probability of getting ace from a deck of 52 cards?

Let $p_S$ be the probability of randomly picking an ace from a set $S$ of $10$ cards. We note that $p_S$ varies only with the number of aces in $S$. More specifically, $$p_S=\frac{\text{nr aces in } S}{10}.$$

The number of aces in $S$ must be in $\{0,1,2,3,4\}$, since there are only $4$ aces in a deck of cards. Hence $p_S \in \{0,1/10,2/10,3/10,4/10\}$.

If we choose $S$ uniformly at random from the set of all $10$-subsets of a deck of $52$ playing cards, then, for $i \in \{0,1,2,3,4\}$, \begin{align*} \mathrm{Pr}(p_S=i/10) &= \mathrm{Pr}(\text{nr aces in } S \text{ is } i) \\ &=\frac{\text{nr 10-subsets with } i \text{ aces}}{\text{total nr 10-subsets}} \\ &= \frac{\binom{4}{i}\binom{48}{10-i}}{\binom{52}{10}} \end{align*} since $S$ comprises of $i$ aces and $10-i$ non-aces.