Arrangements question

Eight women and two men are to sit at a round table. If two men sit directly opposite each other in how many ways can the ten people be arranged?

I know that the answer is 8! = 40320

here is my thinking and i want your thoughts

There is a round table with ten seats around it.
two men sit directly opposite each other. Now I know that the position of the people relative to the table is of no consequence. Thus since the two men sit directly opposite each other, their positions are fixed.
Imagine two men sitting in two seats fixed directly opposite each other.
The only way to count the number of ways is to count the arrangement of the women. This will enable us to consider the position of the people relative to the two fixed men and each other

So, i subtracted 10-2 =8
since there are eight people the arrangement is 8!

Is my reasoning right?

The only thing that you may want to comment on is why you don't consider the two men swapping places. So the answer is not 2*8!

I think the answer to that is since the two people position are fixed then their position relative to the table is of no consequence. We only need concern ourselves with those (mainly the 8 women) who can move and their position depends on the two men

Yeah, I don't think it requires a detailed explanation, but would be worth a mention. If you swapped the men over, you'd have already considered the women's arrangement relative to the new men position, so you're ok.