Coins

a + b + c + d = 6, d ≤ 3, c ≤ 4 C(9, 6) ways

When d > 3, C(5, 2) ways When c > 4, C(4, 1) ways

So, total = 84 - 14 = 70 ways

You want the coefficient of $x^6$

$\left(\sum _{n=0}^8 x^n\right) \left(\sum _{n=0}^7 x^n\right) \left(\sum _{n=0}^4 x^n\right) \left( \sum _{n=0}^3 x^n\right)$

the following table gives you the answer let no. of aluminium coins drawn in a draw = A no. of copper coins drawn=c no. of silver coins=s no.of gold coins=g

we have a+C+s+g=6

a c s g 1 4 1 0\ 1 4 0 1\ 1 3 2 0\ 1 3 1 1\ 1 3 0 2\ 1 2 3 0\ 1 2 2 1\ 1 2 1 2\ 1 2 0 3\ 1 1 4 0\ 1 1 3 1\ 1 1 2 2\ 1 1 1 3\ 1 1 0 4\ 1 0 5 0\ 1 0 4 1\ 1 0 3 2\ 1 0 2 3\ 1 0 1 4\ 1 0 0 5\

0 4 2 0\ 0 4 1 1\ 0 4 0 2\ 0 3 3 0\ 0 3 2 1\ 0 3 1 2\ 0 3 0 3\ 0 2 4 0\ 0 2 3 1\ 0 2 2 2\ 0 2 1 3\ 0 2 0 4\ 0 1 5 0\ 0 1 4 1\ 0 1 3 2\ 0 1 2 3\ 0 1 1 4\ 0 1 0 5\ 0 0 6 0\ 0 0 5 1\ 0 0 4 2\ 0 0 3 3\ 0 0 2 4\ 0 0 1 5\ 0 0 0 6\ 2 4 0 0\ 2 3 1 0\ 2 3 0 1\ 2 2 2 0\ 2 2 1 0\ 2 2 0 2\ 2 1 3 0\ 2 1 2 1\ 2 1 1 2\ 2 1 0 3\ 2 0 4 0\ 2 0 3 1\ 2 0 2 2\ 2 0 1 3\ 2 0 0 4\ 3 3 0 0\ 3 2 1 0\ 3 2 0 1\ 3 1 2 0\ 3 1 1 1\ 3 1 0 2\ 3 0 3 0\ 3 0 2 1\ 3 0 1 2\ 3 0 0 3\