Let's consider all of the possible arrangements for words that DO NOT have the x-letter in it.
Start by drawing a line for the five available slots:
_ _ _ _ _
These will represent the different arrangements of letters.
Since letters may be repeated, these are independent events, meaning the previous does not affect the concurrent term.
Thus, we can have 25 of the 26 different letters, since we're not counting the x.
The second slot can have 25, as well as the third, fourth, and fifth.
We can simplify this to become:
[tex]\text{Arrangements(no x): } 25^{5}[/tex]
Now, let's consider all of the ways in which a five letter word can be made. Using the same logic, we get:
[tex\text{Arrangements: } 26^{5}[/tex]
Thus, the total number of ways in which we can arrange a 5-letter word with at least one X in it is:
[tex]\text{Total arrangements: } 26^{5} - 25^{5}[/tex]
The calculator should do the rest.
