How many strings of six lowercase letters from the English alphabet contain the letters *a* and *b* with all letters different?

Question

How many strings of six lowercase letters from the English alphabet contain the letters a and b with all letters different? Answer: 7650720

I saw a resolution that considered the two types of strings different:

1) Possible strings not containing a nor b.

2) Possible strings not containing a and b.

But I do not understand why. Can someone explain or present another solution?

Answer 1

Here’s one solution: there are $6$ choices for the position of a, and then $5$ choices for the position of $b$. After this we have $24$ remaining possibilities for the next letter (because all letters must be distinct, and we have used two letters already), and then $23$ possibilities, and so on... We then apply the rule of product to give

$$6 \cdot 5 \cdot 24 \cdot 23 \cdot 22 \cdot 21 = 7650720$$

To address your other question, consider the string ‘xya’. It is a string ‘not containing a and b’ because it doesn’t contain both a and b, but it is not a string ‘containing a nor b’ because it contains a.

Answer 2

Another solution:

• First, calculate how many sets of 6 different letters that include a and b there are. The two letters are fixed so we have 24 remaining letters at our disposal and we need to pick 4. It is possible in $\binom{24}{4}$ ways.
• Second, calculate the number of permutations of each such set. Clearly, this is $6!$.
• Finally, multiply the two numbers to get the total:

$$\binom{24}{4} \cdot 6! = 7650720$$