Answer
The number of ways to take different collections of 4 books from 12 books is 495.
Work Step by Step
The order in which collections of 4 books are selected does not matter. Thus, this is a problem of selecting 4 books from a group of 12 books. We need to find the number of combinations of 12 things taken 4 at a time.
We use the formula,
${}_{n}{{C}_{r}}=\frac{n!}{r!\left( n-r \right)!}$
Where $ n=12,r=4$
$\begin{align}
& {}_{12}{{C}_{4}}=\frac{12!}{\left( 12-4 \right)!4!} \\
& =\frac{12!}{8!\times 4!}
\end{align}$
Solve further,
$\begin{align}
& {}_{12}{{C}_{4}}=\frac{12\times 11\times 10\times 9\times 8!}{4\times 3\times 2\times 1\times 8!} \\
& =11\times 5\times 9 \\
& =495
\end{align}$
Thus, there are 495 ways to take different collections of 4 books from 12 books.