I have a probability/combinatorics question that is giving me trouble specifically I don't understand why there is a need for the denominator $6$ "the different orderings of $3$ people" was the explanation but I'm not sure why that is necessary. I've put the question below and my thought process as well.
Question:
If a committee of $3$ people is to be selected from among $5$ married couples so that the committee does not include two people who are married to each other, how many such committees are possible?
Thought Process:
First person of the $3$ selected can be any one among the $10$ folks $$\frac{10!}{(1!)(9!)}$$
Second Person of the $3$ selected must not be married to the first person so lowers availability to $8$ folks to choose from $$\frac{8!}{((1!)(7!)}$$
Third Person of the $3$ selected must not be married to either the first or second person - lowers availability to $6$ folks to choose from $$\frac{6!}{(1!)(5!)}$$
Number of possible ways a committee can be formed: $10 \times 8 \times 6 = 480$
***So I'm able to get to the $480$ but the answer explanation says that I need to divide this $480$ by $6$ and I don't know why because it sounds illogical to do so. Any help would be amazing.