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How many different subsets of a $10$-element set are there where the subsets have at most than $9$ elements?

I know there are $2^{10}$ total number of $10$-element sets.

Please explain to me this problem.

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Let $P(A)$ be the power set of a set $A$ of $10$ elements.

The only element of $P(A)$ with $10$ element is the set consist of $10$ elements.

Hence there are $2^{10}-1$ subset with at most $9$ elements.

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  • $\begingroup$ But we looking for the set that has at most 9 elements, not 10 elements $\endgroup$ – socrates May 24 '17 at 1:03
  • $\begingroup$ @socrates in subsets, order does not matter. Therefore there is only one set that contains all ten elements. $\endgroup$ – John Lou May 24 '17 at 1:04
  • $\begingroup$ @socrates You can either try adding up all the sets that have at most 9 elements, or you can consider all possible subsets, and throw out the ones that have more than 9 elements ... The latter method is far easier, since there is only 1 subset that has more than 9 elements: the set itself! $\endgroup$ – Bram28 May 24 '17 at 1:14

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