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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 as cdb

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

Quiz on Factorials, Permutations and Combinations

1. In how many different ways can you choose a committee of 5 members from a group of 20 people?

The above question is asking how many ways you can pick 5 things from 20 things, which in essence is asking how many combinations of 5 things you can pick from a pool of 20 things i.e.

15504

2. If you flip a coin 10 times, there are 1024 (or 210) possible outcomes. How many of these outcomes have 6 tails?

When you flip a coin once, there are two possible outcomes; a head or a tail. If you flip the coin more than once, the out comes appear in combinations of heads and tails: for example: if you flip the coin twice you'll end up with; 2 heads, or 2 tails, or a head and a tail or a tail and a head. In other words, we're looking for combinations! Therefore the question is asking for

210

3. In how many different ways is it possible to arrange the letters of the word COMBINATIONS?

This question is really simple, the trick is to ignore that misleading choice of word combinations. This question is about permutations since we've been asked to arrange the letters without any order in mind.

All we have to do here is count the number of letters in the word 'COMBINATIONS'

479001600

3b. In how many different ways is it possible to arrange the letters of the word COMBINATIONS with the first 3 letters as BAN?

This question is similar to the one above, we're still being asked for permutations (arrangement) of the letters of the word 'COMBINATIONS'. The only difference here is that we have been asked that the first 3 letters of all the different permutations must be 'BAN'.

So how do we deal with that?

The solution is to subtract the number of letters whose position is constant and then permutate the remaining letters:

Therefore the number of different ways to arrange the letters in 'COMBINATIONS' with 'BAN' as the first letter:

362880

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