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 occurrence 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. $$128^5-127^5$$.

My real question is what is wrong with the reasoning that we have 5 choices for where to put the '@' symbol multiplied by the $$128^4$$ possible strings from the other 4 characters. I can see that $$128^5-127^5 \not= 5(128^4)$$, 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 occurrence 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. $$128^5-127^5$$.

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