Question

You have 12 books and you want to put 8 on a shelf in your room. Order does not matter to you. How many ways can you put 8 books on the shelf?

Answer 1

To find the number of ways to choose 8 books from 12 books where the order does not matter, we can use the combination formula. The number of combinations of \( n \) items taken \( r \) at a time is given by:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

In this scenario, \( n = 12 \) and \( r = 8 \). We need to calculate \( \binom{12}{8} \).

Using the combination formula:

\[
\binom{12}{8} = \frac{12!}{8!(12-8)!} = \frac{12!}{8! \cdot 4!}
\]

We can simplify this expression. The calculation for \( 12! \) can be done as follows:

\[
12! = 12 \times 11 \times 10 \times 9 \times 8!
\]

Now substituting this back into our combination formula:

\[
\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9 \times 8!}{8! \cdot 4!}
\]

The \( 8! \) terms cancel out:

\[
\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4!}
\]

Now, we compute \( 4! \):

\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

So our expression becomes:

\[
\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{24}
\]

Next, we calculate the numerator:

\[
12 \times 11 = 132
\]
\[
132 \times 10 = 1320
\]
\[
1320 \times 9 = 11880
\]

Now, divide by \( 24 \):

\[
\frac{11880}{24} = 495
\]

Thus, the number of ways to choose 8 books from 12 is

\[
\boxed{495}
\]