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?

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My answer is 24. I just multiply 6 and 24 but I am not sure with that answer. I am confused because there are two persons involved.

That's incorrect. Twenty-four would be the number of ways that one person can enter one of four gates and exit one of six exits.

The entrance part is easy. It's just how many ways can the two people enter the same of four different entrances, and that is four, just the number of entrances.

The exit part a little trickier, but not too bad. It's just how many ways can the two people exit in different of six gates. The first person (doesn't matter which is first) can choose from six gates. The second person can choose from the remaining five gates. So there are six times five, i.e. thirty, different ways the two people can exit.

Now the answer isn't thirty, that's just the number of ways two people can exit from two of six different exits. Try to figure out the rest, and I'll tell you if it's right. And show some effort on your other problems and I'll help you with those too.