Original subtitles

Instructor: Hello again.

This is going to be a short lecture

where we focus on the symmetry of combinations.

Unlike permutations and variations,

picking more elements can lead to having fewer combinations.

Imagine you are going on a picnic

and have six pieces of fruit you want to take with you.

However, your basket only has room for four of them.

Using the combinations formula,

we are going to have 15 possible choices.

Therefore, you go out and buy a bigger basket,

but it turns out it can only fit five pieces of fruit.

According to the formula, there are just six ways

of picking the five fruits.

What if you get an even bigger basket

which can fit six fruits?

Well, in how many ways can you pick

six fruits out of six fruits?

Only one, picking all of them.

So we see that, in this case, picking more elements

leads to having fewer combinations.

This is because we can construct the question

in a different way.

Instead of picking which pieces of fruit to take,

we can choose the pieces to leave behind.

Therefore, picking four fruits out of six

is the same as choosing two fruits that will be left out.

Mathematically, selecting two elements out of six

equals six factorial over two factorial

times four factorial, or 15.

How about five out of six fruits?

It's the same as choosing which one

out of the six to leave behind.

In the general case, we can pick P many elements

in as many ways as we can pick N minus P many elements.

This shows us that when it comes to combinations,

the number of possible ways

in which P many elements can be selected is symmetric

with respect to n over 2

And this symmetry is the topic of this lesson.

So far, so good.

Recall the example where we had to select

3 of our 10 employees to represent the company

at a conference.

As we already showed, there are 120 possible selections.

What if instead of choosing three,

we had to pick seven people to go to the conference?

While using the formula we defined in the last lecture,

the number of combinations we would have is 10 factorial

over 7 factorial times 3 factorial.

That's equivalent to 8 times 9 times 10

over 1 times 2 times 3,

or 720 divided by 6

which is 120.

Thus, we would also have 120 different ways

of picking the seven employees.

That's because picking 7 out of 10 employees

to take to the conference

is the same as choosing 3 out of 10 to leave behind.

Great. Please go ahead and check the additional resources

for this course to see how we prove symmetry mathematically.

I believe this would reinforce

your understanding of the topic.

All right.

To sum up, when P is greater than n over 2,

n minus p would be smaller than p.

In such instances, we could apply symmetry

to avoid calculating factorials of large numbers.

Generally, we use symmetry to simplify the calculations

we need to make.

Thanks for watching.