How Many 5-Character Strings Include At Least One '@' Symbol?

[duke_nemmerle]

Homework Statement

Given that the ASCII character system has 128 possible characters how many 5 character strings are there with at least one occurence of the '@' symbol.

Homework Equations

The Attempt at a Solution

So clearly which symbol we're using doesn't matter, and I see that the right answer comes from considering the total number of length 5 character strings minus the number of strings not containing the '@' character i.e. [tex]128^5-127^5[/tex].

My real question is what is wrong with the reasoning that we have 5 choices for where to put the '@' symbol multiplied by the [tex]128^4[/tex] possible strings from the other 4 characters. I can see that [tex]128^5-127^5 \not= 5(128^4)[/tex], but I'm wondering where the flaw in the reasoning is

 

[duke_nemmerle]

Homework Statement

Given that the ASCII character system has 128 possible characters how many 5 character strings are there with at least one occurence of the '@' symbol.

Homework Equations

The Attempt at a Solution

So clearly which symbol we're using doesn't matter, and I see that the right answer comes from considering the total number of length 5 character strings minus the number of strings not containing the '@' character i.e. [tex]128^5-127^5[/tex].

My real question is what is wrong with the reasoning that we have 5 choices for where to put the '@' symbol multiplied by the [tex]128^4[/tex] possible strings from the other 4 characters. I can see that [tex]128^5-127^5 \not= 5(128^4)[/tex], but I'm wondering where the flaw in the reasoning is

I think I've figured it out by considering a much smaller alphabet and character length. I was obviously overcounting, but now I see that I'm counting the strings with more than one instance of the character several times each.

 

[Architeuthis Dux]

.I can provide a response to this question by first acknowledging that your reasoning is not entirely incorrect. In fact, it is a common approach in combinatorics to consider the total number of possibilities minus the number of unwanted possibilities. However, in this case, the flaw in your reasoning lies in the fact that you are overcounting the number of strings with at least one occurrence of the '@' symbol.

Let's break down your approach: you are correct in saying that there are 5 positions where the '@' symbol can be placed within a 5-character string. However, for each of these 5 positions, there are not 128^4 possible combinations of the other 4 characters. This is because, in some cases, the '@' symbol may appear more than once in the string, and your approach does not account for this possibility.

For example, let's say we have the string "a@bc@" and we are considering the position of the first '@' symbol. According to your approach, there are 128^4 possible combinations for the remaining 4 characters, but this does not account for the fact that the second '@' symbol can also appear in any of these 128^4 combinations. Therefore, you are double counting the number of strings with multiple occurrences of the '@' symbol.

In order to correctly count the number of strings with at least one occurrence of the '@' symbol, we can use the principle of inclusion-exclusion. This means we first count the total number of 5-character strings (128^5), then subtract the number of strings with no '@' symbol (127^5), and finally add back the number of strings with two or more '@' symbols (126^5). This accounts for all possible combinations and gives us the correct answer of 128^5 - 127^5 + 126^5.

I hope this explanation helps to clarify the flaw in your reasoning and the correct approach to counting strings with at least one occurrence of a specific symbol. As a scientist, it is important to critically evaluate our methods and make sure we are not overcounting or undercounting our data.