We're only considering lowercase letters, repetition is allowed.
Number of strings of length $4 = 26^4$
Number of strings of length $4$ other than $x = 25^4$
$26^4-25^4 = 66,351$ strings.
This is one solution. But i was thinking of this problem as...
We have $4$ possible positions for $x$. After $x$ is placed, there are $3$ places left. And we have $26^3$ possibilities for those positions.
So, strings that have letter $x = 4\cdot (26^3) = 70304$ strings.
What is wrong with this approach?
You will have to make $4$ cases:-
So, total number of combinations combinations are, $4\cdot25^3+$$4\choose2$$\cdot25^2+$$4\choose3$$\cdot25^1+1=66351$
What is wrong is overcounting, as JimmyK4542 pointed out. You can try to remedy that by inclusion-exclusion, and it works fairly well here.
You've got sets $A_1,A_2,A_3,A_4$ of strings with $x$ in position $1,2,3,4$, respectively, each with $26^3$ elements, and you need to count the elements of their union $A_1\cup A_2\cup A_3\cup A_4$. A first approximation is $|A_1|+|A_2|+|A_3|+|A_4|$, but this counts every string as often as the number of letters $x$ it contains. Strings that contain at least two copies of $x$ lie in some intersection $A_i\cap A_j$ with $i<j$, so one can get the count straight for words with two copies of $x$ by subtracting $|A_i\cap A_j|$ for every such pair $(i,j)$ (there are $\binom42=6$ such pairs). One easily sees that $|A_i\cap A_j|=26^2$ for any such pair. But now the count is off for strings that contain three letters $x$: they have been counted $3$ times initially, but then counted negatively $3$ times (once for each of the $\binom32=3$ pairs $(i,j)$ for which it contributed. A remedy is adding back $|A_i\cap A_j\cap A_k|=26$ for every $i<j<k$. Now the count is right for every string except $xxxx$, which has been counted $\binom41-\binom42+\binom43=2$ times, so we must subtract $1$ to get its count right. It is a general fact that intersections of $m$ distinct sets $A_i$ should contribute with sign $(-1)^{m-1}$. $$\begin{align} |A_1\cup A_2\cup A_3\cup A_4| &=\sum_i|A_i|-\sum_{i<j}|A_i\cap A_j|+\sum_{i<j<k}|A_i\cap A_j\cap A_j|-|A_1\cap A_2\cap A_3\cap A_4|\\ &=\sum_{m=1}^4(-1)^{m-1}\binom4m26^m\\ &=4\times17576-6\times767+4\times26-1\times1 = 66351. \end{align} $$ You can recognise the second line as the binomial formula for $(26-1)^4$, except that the sign is opposite and the initial term $1\times 26^4$ is missing; so your result should be equal to $26^4-(26-1)^4=26^4-25^4$, which indeed it is by the argument initially given in your question.
First: How many ways to have a string of length $4$? $26^4$
Next: How many ways to have a string of length $4$ that does NOT contain $x$? $25^4$
How many ways to have a string of length $4$ that contains the letter $x$? The difference.
$(26^4) - (25^4) = 66351$
Your approach double counts strings like "xxyy", "xyax", "bcxx", etc. which have an x in more than one position. Hence, you end up with a larger result than the actual answer.
