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This would be 128, I believe. Is this correct?

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Yes, I believe it is. For each subset it can either contain or not contain an element. For each element, there are $2$ possibilities. Multiplying these together we get $2^7$ or $128$ subsets.

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For generalisation the total number of subsets of a set containing n elements is 2 to the power n. And you are surely right.

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