Permutations – Outschool
Permutations Lesson
Permutation is a fancy way of saying “arrangement.” With permutations, the ORDER MATTERS.
Ex. How many permutations of 1st, 2nd, and 3rd place finishers are possible from a race with 10 runners?
This is not that difficult. There are 10 options for the first place runner. After someone has won 1st place, there are 9 options for 2nd place and then 8 options for 3rd place.
10 options • 9 options • 8 options
→ 720
There are 720 different permutations possible for the 1st, 2nd, and 3rd places.
This can also be written as the permutation of 10 runners where 3 places are awarded (10P3), where P represents permutation.
The permutation of n things taken r at a time is [latex]_nP_r =frac{n!}{(n-r)!}[/latex] .
Let’s apply the formula to this example. There are ten runners, and we are looking for possible permutations for the top 3 finishers.
[latex]_{10}P_{3} = frac{10!}{(10-3)!}[/latex]
→ [latex]frac{10!}{7!}[/latex]
→ [latex]frac{10 cdot 9 cdot 8 cdot 7 cdot 6 cdot 5 cdot 4 cdot 3 cdot 2 cdot 1}{7 cdot 6 cdot 5 cdot 4 cdot 3 cdot 2 cdot 1}[/latex]
→ [latex]10 cdot 9 cdot 8 [/latex]
→ [latex] 720 [/latex]
There are 720 different possible permutations for the 1st, 2nd, and 3rd places.