# Permutations

A Permutation of a larger set “S” of n objects is any ordered subset of that set. The total number of permutations of r objects from n is denoted $^n P_r$ and a formula for what that is is actually pretty easy to find, if you wanted to think a lil bit.

Just FYI, S={1,2,3} or something, then {1,2}≠ {2,1} as they′re permutations

Using something called the Multiplication Principle, we find that the number of possible outcomes of a multi-stage process is the product of the number of options at each stage.

Consider now a situation where (like in permutations) the order of elements actually matters, so like a race. Now we want to know, if there are n people in the race, how many different permutations of 3 are there? We want to know, so we can consider the probability that racer A, racer B, and racer C get first, second, and third place respectively, so we can tell our relatives how crazy our crazy gamble really was.
This is just a three-stage process where the number of options at each stage is given by the number left in the set to choose from. Some people consider this a “process without replacement”, like, when we consider one person for first place, we don’t let him back in, otherwise he might snag second place too!

So from n objects, the number of permutations of r is given by:

${^n P_r} = n*{(n-1) }*...*{(n-r+1)}= {\frac{n!}{n-r!}}$