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# Permutations

Probability and Statistics
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Probability Theory
schedule Jul 1, 2022
Last updated
local_offer Arthur
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Permutations refers to the number of ways of ordering $r$ elements from a total of $n$ elements.

The formula for permutations is as follows:

$$_nP_r = \frac{n!}{(n-r)!}$$

where:

$n$: total number of elements

$r$: number of elements chosen

# Examples

## Ordering 6 balls

If we had 6 balls numbered from 1 to 6, how many different ways could we order the 6 balls?

Here we are interested in ordering 6 balls from a total of 6 balls.

For the first ball, we can choose any of the 6 balls.

For the second ball, given we have already chosen 1 ball, we only have 5 balls to choose from.

For the last ball, given there is only 1 ball remaining, we only have 1 choice.

Hence this gives us:

\begin{align*} _6P_6 &= \frac{6!}{(6-6)!} \\ &= 6!\\ &= 6 * 5 * 4 * 3 * 2 *1 \\ &= 720 \end{align*}

So we have 720 different ways to order the 6 balls.

## Ordering 3 balls from a total of 6

If we had 6 balls numbered from 1 to 6, how many different ways could we order exactly three of them?

Here we are interested in ordering three balls from a total of 6 balls.

For the first ball, we can choose any of the 6 balls.

For the second ball, given we have already chosen 1 ball, we only have 5 balls to choose from.

For the third and final ball, given we have already chosen 2 balls, we only have 4 balls left to choose from.

Using the formula this gives us:

\begin{align*} _6P_3&=\frac{6!}{(6-3)!}\\ &=\frac{6!}{3!}\\ &=\frac{6*5*4*3*2*1}{3*2*1}\\ &= 6*5*4 \\ &= 120 \end{align*}
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