Probability for 3 of a kind 
How many ways are there to get a three of a kind in 5 card poker?

In other words, out of 52 cards, I need to pick any 5 cards. What is the probability to get the 5 cards of form AAABC where A, B and C are distinct?
Here's my solution but I'm getting an error -
Number of ways to pick the first A card = 52, second A card = 3 and the third a card = 2.
Number of ways to pick the B card = 48
Number of ways to pick the C card = 44
Number of ways to pick 5 cards = $52 \cdot 51 \cdot 50 \cdot 49 \cdot 48$
Therefore the answer is
$$\frac{52 \cdot 3 \cdot 2 \cdot 48 \cdot 44}{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48} = \frac{44}{20825}$$
But the correct answer is $\frac{88}{4165}$.
Where's the error? What's wrong with dividing permutation of success by total permutation instead of combinations of success by total combination?

You have the probability that the first three cards you draw will be the same and the next two both different. $~$ However, the order of drawing the cards should not matter. $~$ Hence you should multiply your result by $\frac{5!}{3!2!}$, the count of ways to select places for the three of a kind.
