Hello! I found some additional information about Combinations and also some tips for Permutations.

Combinations and Permutations

What's the Difference?

In English we use the word "combination" loosely, without thinking if the order of things is important. In other words:

	*"My fruit salad is a combination of apples, grapes and bananas"* We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

	*"The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2*.

So, in Mathematics we use more precise language:

	If the order doesn't matter, it is a Combination.
	If the order does matter it is a Permutation.

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

Combinations

There are also two types of combinations (remember the order does not matter now):

Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so I will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if you have the lucky numbers (no matter what order) you win!

The easiest way to explain it is to:

assume that the order does matter (ie permutations),
then alter it so the order does not matter.

Going back to our pool ball example, let us say that you just want to know which 3 pool balls were chosen, not the order.

We already know that 3 out of 16 gave us 3,360 permutations.

But many of those will be the same to us now, because we don't care what order!

For example, let us say balls 1, 2 and 3 were chosen. These are the possibilites:

Order does matter	Order doesn't matter
1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1	1 2 3

So, the permutations will have 6 times as many possibilites.

In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

3! = 3 × 2 × 1 = 6

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)

So, all we need to do is adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in the order any more):