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 $$b$$.

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 $$a$$.

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).

• Still waking up. Thanks. – N. F. Taussig Sep 17 at 8:28

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.