In how many different ways can six of ten people be seated

Question

in a row of six chairs?

I don't have the answer, but from my rough understanding of permutations and combinations, I got:

10C6 * 6! = 210 * 6!

I figure first 6 people have to be chosen out of the 10, and that can be done in 210 ways. Then, they can be seated in 6! different ways, so I multiplied the two quantities.

Thanks for any help!