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7 classmates could not agree on who would stand in the group photo along with the teacher for the yearbook. How many possible groups can be made such that there is at least one student with the teacher in the photo?

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factorials are not involved in this question at all; we don’t care about how the students are arranged — all we care about is what students are *in* the photo

suppose we *must* have the teacher in the picture and each of the students is either in the picture or out of the picture; then there are 2 possibilities for each student’s “in-picture-ness”

for 7 students, that’s a total of 2^7 =128 different groups that are possible; but we have to subtract 1 from this total, as the case where none of the students are in the photo is disallowed by the question statement

hence the answer is 127

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