Original subtitles

Narrator: Hello, everyone.

The purpose of this video is

to briefly summarize what we learned about combinatorics

in this part of the course.

Okay, we use permutations with variations

when we must arrange a set of objects.

In such cases, the order in which we pick them is crucial.

The major difference between the two is

that, in permutations,

you always arrange the entire set of elements

in the sample space.

For instance, we would use permutations

when we need to arrange the four runners

we have already chosen for our relay.

We have four runners and four positions.

So we rely on permutations.

If, however, we had to pick four out of six people

on the team

and then decide who runs which leg,

we would require using variations.

Alternatively, if we only care

about which four out of the six runners made it

into the team,

we would be dealing with combinations.

In this instance, we do not care who runs which leg.

So order is irrelevant.

Perfect.

Another important topic we discussed is

that there are two types of variations and combinations,

with and without repetition.

When we explore those without repetition,

we see a clear relationship between permutations,

variations, and combinations.

The number of combinations equals the number of variations

divided by the number of permutations.

That is because we count

all the permutations of a given set of numbers

as a single combination,

but as separate variations.

We also define formulas we use to compute their values.

P = n!

V = n!/(n - p)!

And C = n!/p! x (n - p)!.

Recall that these formulas change

when we include repeating values.

V-bar = n to the power of p.

And C-bar = (n + p - 1)!/p!(n-1)!

Great.

Furthermore, we talked about how combinations are symmetric.

Choosing p-many elements out of a set of n-many elements

can be done in the exact same way

as choosing (n - p) many elements.

The reason is we can reverse the problem

and actually choose the elements to be omitted.

So far, we've studied

a significant amount of probability theory.

In the next section,

we will deepen our knowledge

by learning how to deal with multiple events,

introducing you to Bayesian Notation,

independent and dependent events,

and how to compute the probabilities for these events.

See you there

and thanks for watching.