a jury of people is chosen from 10 men and 12 women. How many ways can a jury of 6 men and women be chosen?

Answer 1

You said "How many ways can a jury of 6 men and women be chosen?"
I am sure you meant to say
"How many ways can a jury of 6 men and 6 women be chosen?"

That would be C(12,6)xC(10,6)=....

If you meant it the way you said it, then it would be more difficult.
Do it with different cases

6 women, no men --> C(12,6)xC(10,0)=924
5 women, 1 man ---> C(12,5)xC(10,1)=7920
.
.
.
1 woman, 5 men --> C(12,1)xC(10,5)=3024
no woman, 1 man--> C(12,0)xC(10,6)=210

Now add them all up.

Answer 2

12608