At a museum, there are 4 entrances and 6 exits. In how many ways can two persons enter together at the same gate but exit at different gates? My answer is 60 because for entrance there are 4 choices and for exits there are 15 choices. 4 multiply by 15 is 60. Is that correct?

Yes, that is correct.
4c1*6c2=4*15=60
Therefore, ans= 60 ways.

I'm sorry but that is not correct. There are fifteen, six choose two, choices if we don't care which friend goes through a gate, but there are thirty choices since order is important. For example, if Sally goes through gate 2 and Jimmy goes through gate 5, that is a different way of exiting, then if Sally goes through gate 5 and Jimmy goes through gate 2. Unless I'm reading the question wrong.