Doubt regarding the number of ways of selecting exactly $3$ of a kind from a deck of cards
I want $3$ of a kind, and the other two should be different from this kind, and also they should be of different kind from each other aswell.
I first select one kind out of the $13$ available to me (say I select ACE) and then select $3$ cards from the $4$ available to me from the chosen kind (Suppose I select the ACE of Clubs, Spades and Hearts) in $13\choose 1$$4\choose 1$ ways
Now here is where my doubt begins:
I need the order two cards to be of different kinds and also different from the already selected kind.
One way I can think of doing this is: First select $2$ kinds available from the remaining $12$ kinds, and then select $1$ card each from the $4$ choices available for each kind in $12 \choose 2$$4 \choose 1$ $4\choose 1$
Another Way I can do this is : First I select $1$ kind of the renaming $12$ kinds and then select $1$ card from the $4$ choices available of this kind, then select another kind from the remaining $11$ and then select $1$ card from the $4$ available of this kind in $12\choose 1$$4\choose 1$$11\choose 1$$4\choose 1$
So the possible ways I am getting is $13\choose 1$$4\choose 1$$12\choose 2$$4\choose 1$$4\choose 1$ or $13\choose 1$$4\choose 1$$12\choose 1$$4\choose 1$$11\choose 1$$4\choose 1$
What is the correct way to go?