Factorials, Permutations and Combinations
Factorials
A factorial is represented by the sign (!). When we encounter n! (known as 'n factorial') we say that a factorial is the product of all the whole numbers between 1 and n, where n must always be positive.
For example

0! is a special case factorial.

This is special because there are no positive numbers less than zero and we defined a factorial as a product of the numbers between n and 1. We say that 0! = 1 by claiming that the product of no numbers is 1. The reasoning and mathematics behind this is complicated and beyond the scope of this page, so let's just accept 0! as equal to 1.
This works out to be mathematically true and allows us to redefine n! as follows:

For example

The above allows us to manipulate factorials and break them up, which is useful in combinations and permutations.
Useful Factorial Properties

The last two properties are important to remember. The factorial sign DOES NOT distribute across addition and subtraction.
Permutations and Combinations
Permutations and Combinations in mathematics both refer to different ways of arranging a given set of variables. Permutations are not strict when it comes to the order of things while Combinations are.
For example; given the letters abc
The Permutations are listed as follows

Combinations on the other hand are considered different, all the above are considered the same since they have the exact same letters only arranged different. In other words, in combination, you can't just rearrange the same letters and then claim to have a completely different combination. Combinations are done differently: Given abc, we can make a number of combinations by taking groups of letters at once, i.e.
In groups of 1 we get

In groups of 2 we get

In groups of 3 we get

From the above, you should see that Combinations are about finding how many ways you can combine the different elements of the given entity.
The notation for Combinations is given as

which means the number of combinations of n items taking r items at a time
For example

means find the number of ways 3 items can be combined, taking 2 at a time, and from the example before, we saw that this was 3.
Another example to further illustrate this is as follows:
Given four letters abcd find

solution:
Remember that the order doesn't matter when it comes to combinations, i.e. bcd is the same as dbc which is also the same ascdb

in other words,

Combinations are also commonly denoted as

and in question in the example above could have been asked as

So it is important to remember that

Now that we've seen what combinations are, let us move on to relating factorials and combinations.
The Combination function can be defined using factorials as follows:

We can prove that this is true using the previous example;

which is the same answer we got before.
Let us return to Permutations, which we defined above and also saw an example of. Permutations are denoted by the following

which means the number of permutations of n items taken r items at a time.
For example; given 3 letters abc find

solution:
Remember that the repetition is allowed in permutations unlike in combinations;

which mean that there are 6 ways, in other words

The Permutation function can also be using factorials:

We can prove the above using the previous example

Which is the same answer as before.
If you take a close look at the formulae for Combinations and Permutations, you will be able to see that the two can be expressed in terms of one another, i.e.

from the above, the following relationship can be derived:

The above can be proved by substituting the formula for permutations into the equation

Which as we already saw is the formula for Combinations.
Examples of Factorials, Permutations and Combinations
Example 1
Evaluate the following without using a calculator

Step 1
We have seen that a relatively big number (like 10 in this example) can be broken down into a product of factorials i.e.

Step 2
We can use the above to evaluate the expression as

Step 3
Since 7! appears both in the numerator and denominator, we can proceed to cancel it out

Step 4

Example 2
Evaluate the following

Step 1
We have already defined the combination notation above as:

Step 2
Therefore, we can just substitute in the above formula

Step 3
The numerator and denominator are equal so we can just cancel them out as

Example 3
Evaluate the following expression

Step 1
The notation above shouldn't be all that unfamiliar if you've gone through the page this entire page. We have seen that

Thus it follows that

Step 2
So as in the previous example, we can just substitute and solve

Step 3 but the following is also true

and we can quickly also see that

Step 4
And so we can substitute the above to make the computation easier

Canceling out equal terms in the numerator and denominator results in

Example 4
Compute the following

Step 1
The notation used above is the permutation notation and it means the following:

Step 2
Thus we can substitute for the variable to obtain:

Step 3

Step 4
3! cancels out to leave the following expression