In a poker hand consisting of 5 cards, find the

A poker hand consists of 5 cards (among which we need to have 3 aces).
In the standard 52-card deck, there are 4 aces, and we must select 3 of them.In order to calculate the number of ways of choosing 3 out of 4 aces, we need to use combinations (because the order is not important). So, we can choose 3 out of 4 aces in
$$(\begin{array}{c}4\\ 3\end{array})=\frac{4!}{3!(4-3)!}=4ways$$
We must now select 2 cards from the remaining 48 cards. (52 cards minus 4 aces). The number of ways to choose these 2 cards among 48 cards is
$$(\begin{array}{c}48\\ 2\end{array})=\frac{48!}{2!(48-2)!}=\frac{48\ast 47}{2}=\frac{2256}{2}=1128ways$$
If the first operation can be performed in 4 ways (the number of ways to choose 3 out of 4 aces), and for each of these ways the second operation can be performed in 1128 ways (the number of ways of choosing the remaining 2 cards), then these 2 operations can be performed together in
4*1128=4512 ways
So, there are 4512 ways to choose 3 aces in a poker hand.
There are 52 cards in total, and there are ways to select 5 of them.
$\left(\begin{array}{c}52\\ 5\end{array}\right)=\frac{52!}{5!(52-5)!}=\frac{52\cdot 51\cdot 50\cdot 49\cdot 48\cdot 47}{5\cdot 4\cdot 3\cdot 2\cdot 1}=2598960ways$
So, there are 2598960 ways for a poker hand.
Hence, the probability of choosing 3 aces in a poker hand is
${\displaystyle P=\frac{Thenumberofwaystochoose3cacesinapokerhand}{Thenumberofwaystochooseapokerhand}=\frac{4512}{2598960}=0.00174}$
Result:
0.00174