Question

How many ways can a jury of four men and three women be selected from twelve men and ten women

Answer 1

There are 59,400 ways this jury can be selected.

Since the events are independent of each other (choosing one does not influence the chance of choosing the others) we multiply the combinations. 

The first combination is $_1_2C_4$, because we are selecting 4 men from a possible 12.

The second combination is $_1_0C_3$, because we are selecting 3 women from a possible 10.

Multiplying these we have:  $_1_2C_4 \times _1_0C_3 \\ \\=\frac{12!}{4!8!} \times \frac{10!}{3!7!} \\ \\=\frac{479001600}{967680} \times \frac{3628800}{30240} \\ \\=495 \times 120 \\ = 59400$