Question

In how many ways can $9$ books be arranged on a shelf?

Answer 1

Determine the number of ways that $\mathbf{9}$ books can be arranged on the shelf:

The Fundamental Counting Principle also known as the multiplication rule of counting, states that, if one event has $m$ possible outcome and event $2$ has $n$ possible outcomes, then there $m·n$ total possible outcomes for the two events together.

There are $9$ ways of arranging the first book on the shelf.

There are $8$ ways of arranging the second book after arranging the first book on the shelf.

There are $7$ ways of arranging the third book after arranging the second book on the shelf.

There are $6$ ways of arranging the last book after arranging the second last book on the shelf.

There are $5$ ways of arranging the first book after arranging the second last book on the shelf.

There are $4$ ways of arranging the second book after arranging the first book on the shelf.

There are $3$ ways of arranging the second book after arranging the first book on the shelf.

There are $2$ ways of arranging the third book after arranging the second book on the shelf.

There are $1$ ways of arranging the last book after arranging the second last book on the shelf.

Multiply all the different ways together.

$9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1=362880$.

Hence, the number of ways that $\mathbf{9}$ books can be arranged on the shelf is $\mathbf{362880}$.