# Question a504d

Sep 8, 2017

10! or 3628800

#### Explanation:

First place can be filled in 10 ways second place in 9 ways third place in 8 ways...etc.

So we have 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 This can be written as 10! ( this is known as 10 factorial).

This idea can be a bit difficult to grasp at first.

These types of problems are called permutations.

The general formula for this is;

(n!)/(( n - r )!)

Where $n$ is the number of objects and $r$ is how many are taken at a time.

So above problem would be:

(10!)/((10 - 10 )!) = (10!)/((0)!)

Note: 0! = 1

So we have: 10! = 3628800

Hope this helps you.

Sep 8, 2017

$3 , 628 , 800$ different ways.

#### Explanation:

There are $10$ different possibilities for the 1 st runner.

For each of these $10$ possibilities there are $9$ different possibilities for the 2nd place runner. That is there are $10 \times 9 = 90$ possibilities for the first 2 runners.

Similarly for each of these $90$ possibilities there are $8$ different possibilities for the 3rd place runner. That is there are $10 \times 9 \times 8 = 720$ possible combinations for the first 3 runners.

Similarly there are $10 \times 9 \times 8 \times 7$ possible combinations for the first 4 runners; and
there are $10 \times 9 \times 8 \times 7 \times 6$ possible combinations for the first 5 runners; and
thethere are $10 \times 9 \times 8 \times 7 \times 6 \times 5$ possible combinations for the first 6 runners; and
there are $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4$ possible combinations for the first 7 runners; and
there are $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3$ possible combinations for the first 8 runners; and
re are $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2$ possible combinations for the first 9 runners; and
there are $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ possible combinations for all 10 runners.

$10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3 , 628 , 800$

Note that this is often written as: 10!# (read "10 factorial").
Most spreadsheets and some calculators have a built-in function for this, so you don't need to do the multiple calculations.