(a) agree

(b) Shouldn't that be 240 instead of 270? I get the same expression:

The number of ways to choose the couple is 6C1. Multiply this by the ways to choose the other two people, which is 10C2 - 5C1. It's 10C2 without restriction, and then you subtract off those that form a couple since this is disallowed. The book answer just expanded 6C1(10C2 - 5C1) by distributive property. (Or maybe they reasoned slightly differently but to the same effect.)

This is wrong because for the last two people you are making order matter, thus you are counting duplicates. In fact you count every second pair of people twice, thus your final answer is twice that of the correct answer.