Permutation problem doesn't make sense

Hi there, I have this problem in my discreet maths text book. The answer is given but my answer was half the value and i don't understand why my answer needs to be double. Any enlightment would be most appreciated :

A group contains 6 men and 6 women. How many ways are there to

arrange these people in a row if the men and women alternate.

My solution :

I see 2 ways that this works, either start with men:

m w m w m w m w m w m w

or start with women:

w m w m w m w m w m w m

there are 6! permutations for the first group and also 6! for the 2nd group . So i get the overall total number of ways to arrange men and women alternating is 6!6! = 518400 but the answer is actually 6!6!2 ,why double the answer??

Many thanks!(Happy)