A committee of $6$ people is to be chosen from $8$ students and $6$ teachers so that it contains at least $3$ students and at least $2$ teachers. What is the number of ways this can be done?
I'm having a lot of trouble with this because normally it is only dealing with $1$ variable: How many ways can $x$ fit in $y$ slots. Whereas this has $2$ variables with $6$ slots.
My take was to do it this way:
$8 \cdot 7 \cdot 6 \cdot 5 = 1800$
$6 \cdot 5$
because this takes up $6$ slots.
$8 \cdot 7 \cdot 6$
$6 \cdot 5 \cdot 4$
which totals $2286$. Where am I going wrong?
P.S. My English is poor, so do correct me.