Words over an alphabet that contains at least of each of a subset of letters

I've been struggling to find how many words of length $n \geq 3$ over alphabet S = $\{a, b, c, d, e, f, g, h\}$ contain at least one of each of the letters $\{a, b, c\}$.

My thought process involved counting the total number of possible strings which is $8^n$ and then using the inclusion exclusion principle to subtract the union of the following sets:

1. Strings of length $n$ formed from $\{d,e,f,g,h\}$ denoted $A_1$
2. Strings of length $n$ formed from $\{d,e,f,g,h,a\}$ denoted $A_2$
3. Strings of length $n$ formed from $\{d,e,f,g,h,a,b\}$ denoted $A_3$
4. Strings of length $n$ formed from $\{d,e,f,g,h,b\}$ denoted $A_4$
5. Strings of length $n$ formed from $\{d,e,f,g,h,b,c\}$ denoted $A_5$
6. Strings of length $n$ formed from $\{d,e,f,g,h,c\}$ denoted $A_6$
7. Strings of length $n$ formed from $\{d,e,f,g,h,a,c\}$ denoted $A_7$

which would then yield $8^n - |A_1 \cup A_2 \cup A_3 \cup A_4 \cup A_5 \cup A_6 \cup A_7|$. Is this reasoning correct?

There are a total of $8^n$ words of length $n$. From these, we must subtract those words that do not contain $a$, $b$, and $c$. Therefore, we must subtract the number of words in which at least one of these letters is missing from the total.
Let $A$, $B$, and $C$ denote, respectively, the set of words that contain $a$, $b$, and $c$. Let $A'$, $B'$, and $C'$ denote, respectively, the complements of $A$, $B$, and $C$. Then the number of bad words is $$|A' \cup B' \cup C'|$$ where $$|A' \cup B' \cup C'| = |A'| + |B'| + |C'| - |A' \cap B'| - |A' \cap C | - |B' \cap C'| + |A' \cap B' \cap C'|$$ Can you take it from here?
• Shouldn't it be intersections in $|A′|+|B′|+|C′|−|A′∪B′|−|A′∪C|−|B′∪C′|+|A′∪B′∪C′|$? – William Oct 31 '17 at 15:20