# How many strings of four letters have x in them

• Mr Davis 97
In summary: That's why you get a different answer.In summary, there are two ways to approach the problem of finding the number of strings of four lowercase letters that have an x in them. The first is to calculate the number of strings with exactly one x, which is 62,500. The second is to calculate the number of strings with at least one x, which is 66,351. The difference in these two numbers is due to the distinction between exactly one x and at least one x.
Mr Davis 97

## Homework Statement

How many strings of four lowercase letters have x in them?

## The Attempt at a Solution

It seems that there are two ways of doing this. First, there are four ways to choose where the x goes, and then there are 25^3 number of ways to choose what the other letters are. So in total there would be 4(25)^3 = 62500 ways.

But another valid way would be to count the complement. That is, there are 26^4 strings of four lowercase letters, and there are 25^4 strings of lowercase letters with no x. Thus 26^4 - 25^4 = 66351 would be how many lowercase letters have an x.

These two numbers obviously don't match. So what could I be doing wrong?

Mr Davis 97 said:

## Homework Statement

How many strings of four lowercase letters have x in them?

## The Attempt at a Solution

It seems that there are two ways of doing this. First, there are four ways to choose where the x goes, and then there are 25^3 number of ways to choose what the other letters are. So in total there would be 4(25)^3 = 62500 ways.

But another valid way would be to count the complement. That is, there are 26^4 strings of four lowercase letters, and there are 25^4 strings of lowercase letters with no x. Thus 26^4 - 25^4 = 66351 would be how many lowercase letters have an x.

These two numbers obviously don't match. So what could I be doing wrong?
Is the question "How many strings of four lowercase letters have exactly one x in them?" It makes a difference whether there is exactly one occurrence of this letter or if there can be two or more of these letters in the string.

Mark44 said:
Is the question "How many strings of four lowercase letters have exactly one x in them?" It makes a difference whether there is exactly one occurrence of this letter or if there can be two or more of these letters in the string.
Ah! I read the question wrong. Does this mean that first answer is incorrect, as it deals with the case of having exactly one, while second answer is correct, since it deals with the case of having at least one x?

Mr Davis 97 said:
Ah! I read the question wrong. Does this mean that first answer is incorrect, as it deals with the case of having exactly one, while second answer is correct, since it deals with the case of having at least one x?
The first calculation you did is for exactly one x. There are four ways to choose where the x goes, times the 25 other possibilities in the other three positions. IOW, ##\binom 4 1 \cdot 25^3 = 62,500##. Your other calculation allows for at least one, but no more than four x's to appear.

## 1. How many possible strings of four letters have "x" in them?

The answer depends on the context of the question. If we are considering all possible combinations of four letters in the English alphabet, there are 456 strings that contain the letter "x". However, if we are only considering valid words in the English language, the number significantly decreases to only 121 words.

## 2. How can we calculate the number of four-letter strings with "x"?

To calculate the number of four-letter strings with "x", we need to consider the position of the letter "x" in the string. If "x" is allowed to appear in any position, there are 26 possible ways to choose the first letter, 26 for the second, 26 for the third, and 26 for the fourth. This gives us a total of 26 x 26 x 26 x 26 = 456976 possible strings. However, if "x" must be in the first, second, third, or fourth position, the calculation would be (1 x 26 x 26 x 26) + (26 x 1 x 26 x 26) + (26 x 26 x 1 x 26) + (26 x 26 x 26 x 1) = 456 + 676 + 676 + 676 = 2484 possible strings.

## 3. How many four-letter strings have exactly one "x"?

If we are considering all possible combinations of four letters in the English alphabet, there are 104 strings that have exactly one "x". However, if we are only considering valid words in the English language, the number decreases to only 22 words.

## 4. How many four-letter strings do not contain the letter "x"?

If we are considering all possible combinations of four letters in the English alphabet, there are 17576 strings that do not contain the letter "x". However, if we are only considering valid words in the English language, there are 456 possible strings.

## 5. How many four-letter strings have "x" as the first and last letter?

If we are considering all possible combinations of four letters in the English alphabet, there are 26 possible strings that have "x" as the first and last letter. However, if we are only considering valid words in the English language, there are only 1 possible string - "exam".

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