Let $A$ denote the set of the Latin alphabet. Let $B$ be a subset of $A$ such as $|B|=10$. How to claculate how many such subsets $B$ don't have for a subset the set $\{a, b, c, d\}$. In other words $$ |S| = ? \quad\text{for}\quad S = \{B\mid B\subset A\land |B| = 10\land \{a, b, c, d\}\not\subset B\} $$
My approach so far is to calculate how many such $B$'s exist, which I finded to be $C(26, 10) = 5.311.735$ and substract from them how many $B$'s contain the set $\{a, b, c, d\}$. The thing is I can't find a way to calculate that last quantity.
Any help is appreciated. Thanks in advance!
The number of subsets where we don't care is indeed $\binom{26}{10}$ as you correctly found. Further you are correct that to get the count that you are interested in, we can subtract the number of subsets which do contain all of $a,b,c,d$.
To find the number which do contain all of $a,b,c,d$... we know that each of them have those four letters and the only information missing is which other six letters are in the set out of the $22$ remaining letters. That too is a standard problem solved by binomial coefficients, $\binom{22}{6}$.
$$\binom{26}{10}-\binom{22}{6}$$