A Senate committee has 8 Republicans and 6 Democrats.  In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats?
There are 8 Republicans and 3 spots for them, so there are $\binom{8}{3} = 56$ ways to choose the Republicans.  There are 6 Democrats and 2 spots for them, so there are $\binom{6}{2} = 15$ ways to choose the Democrats.  So there are $56 \times 15 = \boxed{840}$ ways to choose the subcommittee.