Original subtitles

Instructor: Welcome back, everybody.

In this lecture,

we are going to introduce you to combination,

what they are and when to use them.

Combinations represent the number of different ways

we can pick certain elements of a set.

Imagine you were trying to pick three people

to represent your company

on a very important technology related conference.

There are 10 people working in the office,

so how many different combinations are there?

If you calculated this as a variation

your answer would be 720,

but you would be counting every group of three people

several times over.

This is because picking Alex, Sarah, and Dave

to go to the conference

is the same as picking Alex, Dave, and Sarah.

As you just saw, variations don't take into account

double counting elements.

That is where combinations step in.

This next sentence might sound confusing at first

so please pay close attention.

We can say that all the different permutations

of a single combination are different variations.

Let us look at the Sarah, Alex, and Dave example.

Choosing those three to represent the company

is a single combination.

Since the order in which we pick them is not relevant,

choosing Sarah, Alex, and Dave

is exactly the same as choosing Sarah, Dave and Alex,

Dave, Sarah and Alex, Dave, Alex, and Sarah,

Alex, Dave and Sarah or Alex, Sarah, and Dave.

Any of the six permutations we wrote

is a different variation but not a different combination.

That is what we meant when we said that combinations

take into account double counting.

Okay.

Recall that the formula for calculating permutations

of N many elements is simply N factorial.

Since N is three in this case,

there would be a total of six permutations

for choosing Alex, Dave, and Sarah.

Since variations count these six as separate,

we are going to have six variations for any combination.

This means that we are going to end up

with six times fewer combinations than variations.

Using the formulas we already know,

there are 10 times 9 times 8, or 720 variations.

In terms of combinations, we have 720 divided by 6,

or 120 ways of choosing who represents the company.

Woo! Good job.

Let's repeat the sentence we used before once again.

We can say that all the different permutations

of a single combination are different variations.

There are 6 permutations, 120 combinations,

and 720 variations.

Hope it's much clearer now.

Okay.

From here, you can already imagine what the formula is.

Let's construct it together.

What's the number of combinations

for choosing P many elements

out of a sample space of N elements?

As you saw in the last example,

the number of combinations equals the number of variations

over the number of permutations.

Mathematically, we would write that as C equals V over P.

If we plug in the formulas

associated with variations and permutations,

we get the following formula;

N factorial over P factorial times N minus P factorial.

Let's apply the formula to our example.

The number of combinations equals 10 factorial

over 3 factorial times 7 factorial.

After simplifying, we get 8 times 9 times 10

over 1 times 2 times 3.

This is equal to 720 over 6, which is 120.

Exactly what we got earlier.

Just to make sure everything is on the same page,

let's go through another example.

What if we had to choose 4 out of 10 people

to go to the conference?

According to the formula,

we are going to have 10 factorial over 4 factorial

times 6 factorial combinations of doing so.

After some simplifications,

this would equal 7 times 8 times 9 times 10

over 1 times 2 times 3 times 4.

After computing the values this equals 5,040 over 24,

which is 210.

Good job everyone.

In the next section

we are going to introduce

an important property of combinations, symmetry.

See you all there and thanks for watching.