1. The problem statement, all variables and given/known data How many strings of five ASCII characters contain the character @ ("at" sign) at least once? [Note: there are 128 different ASCII characters.] 2. Relevant equations The rule of product and inclusion-exclusion principle are relevant. 3. The attempt at a solution The correct solution is as follows: The number of 5-character ASCII strings is 128^5. The number of 5-character ASCII strings not including at least one @ is 127^5. By the inclusion-exclusion principle, the number of 5-character ASCII strings including at least one @ is equal to 128^5 - 127^5. I have no problem with that. What bothers me is that I can't find out where I go wrong with the following "solution", which yields a different answer. Incorrect solution: At least one of the five characters is an @. There are 5 ways to place this character, because the string has a length of 5. The remaining characters may or may not be an @ symbol. Each of the four remaining characters can be chosen in 128 different ways. By the rule of product, there are 5 * 128 * 128 * 128 * 128 = 5*128^4 such strings. Query: 128^5 - 127^5 is a much larger number than 5*128^4. Which assumption in my incorrect solution is unjustified?