Question

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

Answer 1

Hint: We have to find the number of ways in which we can select at least one of the 12 different books on the shelf. The number of ways to select at least one of the 12 books is to first of all select all the possible ways to select 12 different books then subtract this result with the result of the none of the books selected. Number of ways to select none of the books is 1 not selecting any of the books.

Complete step-by-step answer:
It is given that there are 12 different books in a shelf and we have to find the number of ways to select at least one of the 12 different books.
First of all write all the possible ways to select 12 different books from the shelf. There are two possibilities two select any one of the book whether we can take it or leave it and as there are 12 different books so the possible number of ways are: ${{2}^{12}}$
Now, to find the ways to select at least one of the 12 different books we are going to subtract the above result with the number of ways when none of the books is selected.
There is only 1 way to select none of the book by not selecting any of the books from the shelf. Hence, the ways to select at least one of the 12 different books is as follows-
$\begin{aligned} & {{2}^{12}}-1 \\\ & =4096-1 \\\ & =4095 \\\ \end{aligned}$
Hence, there are 4095 ways to select at least one of the 12 different books from the shelf.

Note: The point where you go wrong in the question is in selecting all the possible ways of selecting 12 different books. You might write the number of ways of selecting 12 different books from 12 different books as:
${}^{12}{{C}_{12}}$
The above selection is wrong because here you are ignoring the condition when we are not selecting a book because all the possible ways contain the possibilities when none of the book is selected.