I agree with the other two about the third term. But I disagree about the answers they came up with.

**The given answer, 21600, is correct.**
There are 6 ways, (AB, AC, BA, BC, CA, CB), for two of the three to sit together.

Now we have in the pattern _D_E_F_G_H_ six places to put one of those pairs and the five places to put the remaining third-person out so that the three are not seated together.

That gives

.

So I think the given answer is correct, but I cannot understand the why of makeup of those three terms.

Post Script

There are

ways to separate A, B, & C.

There are

ways that at least two of those are together. But this number also includes cases in which all three are together.

So There are

.