A box of alphabet blocks contains $6$ of each $A,B,C,D$, $2$ of each of $E,F,G$, $4$ of each of $H,I,J,K,L,M$, and one of each other letter. Find the generating function for selecting $r$ (unordered) blocks from this set.
So I have,
$(1+x+x^2 +x^3 +x^4 )^6 (1+x+x^2 +x^3 )^2 (1+x+x^2 +...+x^6 )^4 (x+x^2 +...+x^{13} )$
Not sure if I did this correctly.
I'm sure you've lost a $1$ from the last bracket.
I think what you have there is the generating function for the problem when you have 4 each of A, B, C, D, E, F, 3 each of G and H, 6 each of I, J, K and L, and 13 Ms.