You're not paying attention to whether any of the four cards you pick from the remaining 48 have the same face value.

You need to pick your five different face values. There are $\binom{13}{5}$ ways to do that.

Then you need to pick which of the five face values will be the three. There are five ways to do that.

Then you need to decide, for each face value of which you only have one card, which suit that card will be. There are four ways to do that for each face value, so $4^4$ ways in all.

Finally, you need to decide which three suits will be your three of a kind. There are $\binom{4}{3}=4$ ways to do that.

So overall, there are $\binom{13}{5} \times 5 \times 4^5 = 1647360$ ways to pick your hand.

You're not paying attention to whether any of the four cards you pick from the remaining 48 have the same face value.

You need to pick your five different face values. There are $\binom{13}{5}$ ways to do that.

Then you need to pick which of the five face values will be the three. There are five ways to do that.

Then you need to decide, for each face value of which you only have one card, which suit that card will be. There are four ways to do that for each face value, so $4^4$ ways in all.

Finally, you need to decide which three suits will be your three of a kind. There are $\binom{4}{3}=4$ ways to do that.

So overall, there are $\binom{13}{5} \times 5 \times 4^5 = 6589440$ ways to pick your hand.