How do I find the total number of arrangements when arranging Jeff, Brittany, Alex, Kelsey, and Zach to jump off the diving board?
This problem is a common one in a probability course. We are trying to count the number of possibilities to arrange people in a certain order. They will all be jumping, so there are no tricks here!
To start, set up the number of positions that each person may take:
____ _____ _____ _____ _____
In the first position, there are 5 people that may volunteer to go first. For each possibility here, there are 4 more people that can take the second position, 3 more for the third position, and so on. This gives us the following relation, where `P` is the number of possibilities:
`P = 5*4*3*2*1 = 5!`
The `!` operator is the "factorial" operator. It simply means we multiply the number by every other number before it until we hit 1. Evaluating the factorial gives us the following result:
`P = 120`
There are many problems like this where only 3 people are jumping or 2 people or what-have-you. In this case, there is a function used in probability called "permutation." It is given by the following formula, where `n` is the number of people and `k` is the number of possible positions:
`_n P_k = (n!)/((n-k)!)`
We can use this formula, too, to calculate the number of ways for people to jump:
`P = _5 P_5 = (5!)/((5-5)!) = (5!)/(0!) = 5! = 120`
Notice that ` 0! = 1`. This relationship is critical to understand many of these sorts of formulas.
Again, our final answer is 120 possible permutations of people diving.