How many proper subsets of ${1,2,3,4,5,6,7}$ contain the numbers $1$ and $7$? | Aspire Study

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How many proper subsets of ${1,2,3,4,5,6,7}$ contain the numbers $1$ and $7$?

1
2
3
4

Solution

Fix $1$ and $7$ in the subset. Remaining elements = ${2,3,4,5,6}$ → $5$ elements. Number of subsets = $2^5 = 32$. Proper subset means whole set is excluded → still $32$.

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