To invite at least one person from 10, subtract the case of inviting nobody: 2^10 - 1 = 1024 - 1 = 1023. Each person has 2 choices (invited or not), giving 2^10 total subsets. Excluding the empty set leaves 1023 valid options.
For a group of 10 people, the number of ways to choose at least one person is the total number of non-empty subsets, which is 2^10 − 1 = 1023 (subtracting the one case of choosing nobody). This combinatorial trick — total subsets minus the empty set — is the standard way to count "at least one" selections.