I tried to approach this problem by using complementary counting. The number of ways to get a 6-letter word with no restriction is $26^6$. Then we can subtract the number of ways where the word does not contain "a, b, c, or d", which is $22^6$.
But then I realized there was a problem with this.
This problem is weird in that the word has to contain "a" and ANY of "b, c, or d." For example, the following would be valid words: "abzzzx", "axkkd", "yuabd". However, it would not be valid if the word is "bcdbcz" or "azzzzz".
I hope this makes sense. I was thinking maybe exclusion-inclusion needs to be used somewhere, but I'm not sure if this thinking is even right.
This is an extension to a counting problem I had in class (I thought of it).