How many strings are there of four lower-case letters that have the letter x in them? All letters can be reused. Here is an incorrect solution. Can you figure out what is wrong?
A four letter string that contains x can be constructed in two steps as follows. In Step 1
we choose one of the four positions for x (4 ways of doing this). In Step 2 we choose three
letters for the remaining three places (263 ways of doing this). By the multiplcation rule,
there are 4 · 263 = 70304 four letter strings that contain x.

The given solution incorrectly assumes that choosing one position for the letter "x" and then selecting three remaining letters from the 26 lowercase letters will give all possible strings with at least one "x." However, this solution actually counts only the cases where exactly one of the letters is "x," while the problem asks for **all** strings that contain at least one "x" (which includes cases with more than one "x").

### Problem with the solution:
The solution is based on the following flawed reasoning:
1. It counts only strings with exactly one "x," but we need to count strings with at least one "x."
2. The proposed solution multiplies the number of ways to place "x" in exactly one position by the number of possible letters for the remaining three positions, which does not account for strings where "x" appears in two or more positions.

### Correct approach:
1. **Total number of strings without any restrictions:** 
   There are 26 possible lowercase letters, and since each of the four positions in the string can be filled by any of the 26 letters, the total number of possible strings is:
   \[
   26^4 = 456976
   \]
2. **Number of strings that contain no "x":**
   If no "x" is allowed, then each of the four positions can only be filled by one of the remaining 25 letters. So, the number of strings without any "x" is:
   \[
   25^4 = 390625
   \]
3. **Number of strings that contain at least one "x":**
   To find this, subtract the number of strings without any "x" from the total number of strings:
   \[
   26^4 - 25^4 = 456976 - 390625 = 66351
   \]

So, the correct number of four-letter strings that contain at least one "x" is **66,351**.

### Do you want details or have any questions?
Here are 5 related questions to expand your understanding:
1. How would the answer change if we were looking for strings of 5 letters instead of 4?
2. What if we required exactly two occurrences of the letter "x"?
3. How does the inclusion-exclusion principle apply to this problem?
4. Can this be solved using binomial coefficients?
5. How many four-letter strings contain at least one vowel?

**Tip:** In combinatorics, the "at least one" approach is often easier when using the complement (subtracting the cases where the condition doesn't hold).

How many strings are there of four lower-case letters that have the letter x in them? All letters can be reused. Here is an incorrect solution. Can you figure out what is wrong?

Explore how to count the number of four-letter strings containing the letter 'x' using the complement principle. Learn how incorrect solutions can be corrected by considering all possible cases. Suitable for Grades 9-12, with key formulas and combinatorial reasoning.

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