Why isn't $26^6 - 24^6$ the number of possible permutations of the alphabet without "a" and "b"? The question is "How many strings of six lower case letters from the English alphabet contain the letters $a$ and $b$?"
Why doesn't $26^6 - 24^6$ work? 
$26^6$ is all the possible permutations of $26$ letters MINUS all permutations without $a$ nor $b$.
 A: You want to count the strings the contain $a$ AND $b$. What you should subtract is thus the number of strings that don’t contain both $a$ and $b$, meaning that they are allowed to contain $a$ but not $b$, and $b$ but not $a$.
A: For your mistake see the answer of Mankind.
This answer tells you how to do it correctly.

Let $A$ denote the collection of strings that do not contain letter $a$ and let $B$ denote the collection of strings that do not contain letter $b$.
Then the number of strings that contain letter $a$ and letter $b$ equals: $$|A^{\complement}\cap B^{\complement}|=26^6-|A\cup B|=26^6-|A|-|B|+|A\cap B|=26^6-2\cdot25^6+24^6$$
Further for completeness observe that: $$|A^{\complement}\cup B^{\complement}|=26^6-|A\cap B|=26^6-24^6$$counting the number of strings that contain letter $a$ or letter $b$ (where it is allowed to contain both).
A: When facing a problem like this solve it for small numbers.
Our alphabet is {a,b,c}.  How many 2 character long strings?  $3^2 = 9$.  What are they?
aa
ab
ac
ba
bb
bc
ca
cb
cc

How many 2 character long strings with a or b?  $1^3 = 1$.  It is ccc.
How many 3 character long strings with both a and b?
ab
ba

2.  Using your method, you would get 9-1=8.  Now, that minus is actually taking a specific one way, so we can see what 8 we counted here:
aa
ab
ac
ba
bb
bc
ca
cb

we can see that we counted every string with either an a or a b, not string that have both.
Now we can count strings that do contain a single letter this way.  If we take away the lines that don't contain a:
aa
ab
ac
ba
bb
bc
ca
cb
cc

minus:
bb
bc
cb
cc

We get $3^2 - (3-1)^2 = 5$:
aa
ab
ac
ba
ca

which is the right answer for the number of 2 character strings that contain a.
