There are $12$ different books in a shelf. In how many ways we can select at least one of them $?$

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Solution

We may select $1$ book, $2$ books, ..., $12$ books.
$∴$ The number of ways $=^{12} C_{1} +^{12} C_{2} + . . . +^{12} C_{12} = 2^{12} - 1 = 4095$

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There are 12 different books in a shelf. In how many ways we can select at least one of them?

We may select 1 book, 2 books, ..., 12 books.∴ The number of ways=12C1+12C2+...+12C12=212−1=4095

There are $12$ different books in a shelf. In how many ways we can select at least one of them $?$

We may select $1$ book, $2$ books, ..., $12$ books.
$∴$ The number of ways $=^{12} C_{1} +^{12} C_{2} + . . . +^{12} C_{12} = 2^{12} - 1 = 4095$