Counting the number of distinct sub-sets of size $x$ selectable from a non-distinct set of size $y$

Question

How do I count the number of selections of size $x$ from a set of $y$ objects where there may be more than $1$ of some object type, where these duplicates are distinguishable but I never have more than $1$ of each.

e.g. If I have a set of $13$ objects:

$A1,B1,C1,D1,E1,F1,G1,H1,H2,I1,I2,I3,I4$

How do I count the number of valid ways to select $6$ objects from this set where I never have more than $1$ object of the same letter, assuming the order doesn't matter and there is no replacement.

e.g.

$A1,B1,C1,D1,E1,F1$ == VALID
$A1,B1,C1,D1,E2,F1$ == VALID
$B1,C1,D1,F1,H2,I4$ == VALID

but

$A1,B1,C1,D1,H2,H3$ == NOT VALID

etc.

Thanks for your input in advance.

Answer 1

Make cases, in this instance none from type $H,I,\;$ one from type $H$ only,$\;$ one from type $I$ only, and one each from types$\;H,I$ to get

$\binom76 + \binom21\binom75 + \binom41\binom75 + \binom21\binom41\binom74$