Permutation is the number of ways a certain number of objects can be arranged in a distinct order. It is the number of ways to arrange n different objects in a distinct order of r number of positions. It be denoted:
40 staffs arranged in 3 positions
Say you have a project group of 40 staffs for which you are going to select and arrange a manager group composed by three of these staffs. The three positions will be arranged between the three positions:
- project manager
- project manager assistant
- project supervisor
In how many ways can the 3 positions be occupied when the order matters?
Say we pull out the three chairs for the three positions. We start by counting how many staffs can be chosen to sit on the first chair. The answer is 40. Then, after the first chair being seated, there are 39 persons remaining to sit on the second chair and 38 for the third chair.
So, the calculation is 40 x 39 x 38 = 59,280, meaning that the 40 staffs can be arranged in 59,280 different ways between these 3 positions.
The general formula for the calculating that we just did is:
The formula says “40 factorial but please stop at 38”. In order to stop at 38, we divide with the remaining 37 saying (40-3)! which is the (n-r)! = 37!
8 Permute 5
In how many ways can 8 friends be arranged to sit in 5 chairs? This would be:
The formula says: How many friends are there to sit on the first chair? There are 8. Then, in how many ways can the remaining friends sit on the second chair (7), and in how many ways can the remaining friends be arranged to sit on the third chair (6).
The answer is: 8x7x6. So, we need a formula to say “8! but please only count 3 down”: 8x7x6. This can be done by (n-r)! which in this case is (8-3)! = 5! Then we get:
10 Permute 4
If you have a group of 10 friends and only 4 chairs, in how many ways con your friends be seated in the 4 chairs? Your 10 friends can be “permuted” in 4 chairs:
Your 10 friends can be seated in 5040 different ways in the 4 chairs.
Permutation vs combination
What if we have a group of 20 soccer players and wish to make a start-out team of 11 players? In this case, we only care for the group of who the players are and not for the number of ways they can be arranged in the different positions on the pitch.
In this case, the order, or the positions, or the arrangements, do not matter. We only care for the group of players. We only care for who they are, not for what positions they can take.
This is no longer permutations, as permutations cares about the order. The permutation formulation would have said “in how many ways can 20 players be arranged to play in 11 different positions”. But the question that we are being asked now is WHO are the players. Not how in how many ways they can be arranged.
In permutation the order matters, in combination order doesn’t matter.
MS Excel for permutations
- Khan Academy video: Factorial and counting seat arrangements
- Khan Academy video: Permutation formula
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