Counting 4-Letter Strings Containing the Letter 'x'

The English alphabet comprises 26 unique lowercase letters. A common problem in combinatorics is to determine how many strings of a specific length, in this case, four lowercase letters, can be formed with at least one 'x'. This article breaks down the process using complementary counting and explores various patterns.

Complementary Counting Method

To find the number of strings of four lowercase letters that contain at least one 'x', we can use the principle of complementary counting. This method involves first calculating the total number of possible strings and then subtracting the number of strings that do not contain 'x'.

Step 1: Calculate the Total Number of Strings

Since there are 26 letters in the alphabet, the total number of possible strings of length 4 is given by (26^4).

26^4 456,976

Step 2: Calculate the Number of Strings Without 'x'

When we exclude 'x', there are 25 letters remaining in the alphabet. Thus, the total number of strings of length 4 that do not contain 'x' is (25^4).

25^4 390,625

Step 3: Calculate the Number of Strings Containing at Least One 'x'

To find the number of strings that contain at least one 'x', we subtract the number of strings without 'x' from the total number of strings.

Number of strings with at least one 'x' 26^4 - 25^4

456,976 - 390,625 66,351

Therefore, the number of strings of four lowercase letters that have the letter 'x' in them is 66,351.

General Patterns

We can also consider the general patterns we could have when forming these strings. Let's explore different placements of 'x' within the four-letter string:

4 patterns with 1 'x': x---, -x--, --x-, ---x 6 patterns with 2 'xs': xx--, x-x-, x--x, -xx-, -x-x, --xx 4 patterns with 3 'xs': xxx-, xx-x, x-xx, -xxx 1 pattern with 4 'xs': xxxx

In each case, the remaining dashes can be filled with any of the 25 other letters in the alphabet. Hence, the number of variations for each pattern can be calculated as:

For patterns with 3 dashes, each has (25^3) variants. For patterns with 2 dashes, each has (25^2) variants. For patterns with 1 dash, each has 25 variants. The pattern with no dashes has just 1 variant.

The total number of strings is then given by:

4 × (25^3) 6 × (25^2) 4 × 25 1

The calculation yields:

4 × 15,625 6 × 625 4 × 25 1 66,351

Permutations with or without Repetition

The total number of 4-letter permutations, including those with repetition of letters, is (26^4). This can be called 'A'. The number of 4-letter permutations without 'x' is (25^4), which can be called 'B'. Therefore, the number of strings that include 'x' is given by `A - B`.

26^4 - 25^4 66,351

From the total possible 4-letter permutations minus the 4-letter permutations that exclude 'x', we get 66,351.

Conclusion

In conclusion, using complementary counting, we can effectively determine that the number of four-letter strings containing the letter 'x' is 66,351. This approach highlights the power of combining combinatorial methods to solve such problems efficiently.