Permutations part three

We come across factorials when working out the number of permutations.

When we worked out the number of ways we could order three coloured cubes into three positions we found the answer was $3 \times 2 \times 1 = 6$ this is the same as $3!$

We also worked out the number of ways we could order five coloured cubes into three positions.

We worked this out to be $5 \times 4 \times 3 = 60$

We can express this using factorials. Notice that

$5 \times 4 \times 3 = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}$

$5 \times 4 \times 3 \times 2 \times 1 = 5!$

$2 \times 1 = 2!$

$5 \times 4 \times 3 = \frac{5!}{2!}$

This is true for all permutations where you need to arrange n objects into r positions. The number of ways to do this is $\frac{n!}{(n-r)!}$

For our example of the number of ways to arrange 5 coloured blocks in three positions, this is $\frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60$

There is a factorial button on scientific calculators and you can use this to find the value of factorials.

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Permutations part three