I see the answers for (a)(ii) and (b)(i) are 870 and 50803200 respectively

(a)(ii)
have to consider 3 different cases
- one member from each couple + 2 others
ways = 2C1 x 2C1 x 10C2 = 180
- one member from one couple + 3 others
ways = 2C1 x 2C1 x 10C3 = 480
- no couple + 4 others
ways = 10C4 = 210
total ways = 180 + 480 + 210 = 870

(b)(i)
since must alternate, then must be
M W M W M W M W M W M W M W
or
W M W M W M W M W M W M W M,
ways = 7! x 7! x 2 = 50803200

Thanks!

For the 2nd case of part (a)(ii), why do you multiply the 2 2C1?

for the 2nd case we are considering one person is selected from among the two couples

1st 2C1 is to select one couple from two couple
2nd 2C1 is to select one person from the one couple.

that is reason to have two 2C1.

… another way you can look at it is that among the 2 couples, there are total of 4 people.
you are selecting one person out of 4,
so you can also calculate it as 4C1 x 10C3 = 480 (same answer)