Six married couples are in a room

How many ways can 4 people be randomly chosen if:

a) The four people chosen represent two married couples

b) Exactly one married couple is among the four people chosen

a) 6C2 = 15 ways.

b) Here is where I became stuck: total choice of 4 people are:

6C1 x 10C2

However, this include arrangements of couples

To get rid of double couples, 6C1 x 10C2 - 6C2 to obtain 255 ways

The book however has 270, by 6C1 x 10C2 - 6C1 x 5C1

Also, when doing b) why is it wrong to do this way:

First select the couple to be in the group. Then multiply this by no. of people available. Then multiply by no. of people left, which does not belong to couple pair.

6C1 x 10 x 8