Let's consider the following problem.

You have an unlimited supply of tomatoes and bell peppers and lettuce.

We want to make a salad out of four units and one of these three ingredients.

You don't have to use all three ingredients.

You just have to pick four units among three options.

How many different salads- salads we can make?

Let's see what we have in this problem.

We pick for items out of three options with repetitions.

And the order doesn't matter,

if we pick items in the different order,

then this is still the same salad.

So this is our second combinations with repetitions order doesn't matter.

We allow repetitions of options.

We still do not know how to compute the answer immediately.

We don't have a formula.

What we will do in this example we will list all possible salads.

And then we will count them.

But we would like to do it wisely.

We would like to build up someone to each and for

the future solution of this problem in general.

So let's look at the picture.

You will know tomatoes by red circles,

bell pepper by yellow circles,

and lettuce by green circles.

So we have four ingredients and, for example,

we can pick a salad like this or we

can pick a salad like this and not that these are just the same salads.

Order doesn't matter. We have two units of

tomatoes one unit of bell pepper and one unit of lettuce.

Since our order doesn't matter.

Let's do the following.

Let's draw a picture of tomatoes first,

then bell peppers, then lettuce and let's do this always.

So which way we will not count the same thing twice?

This is the same salad but now we have ordered it.

Okay, now we have to list all possible salads

so you have to do it in some ordered way, some reasonable way.

Let's do the following.

Let's consider all possible numbers of tomatoes in the salad.

Let's consider all possible cases

and we will count the number of salads in each case separately.

Okay let's proceed to this.

Here's our salad. And let's start with the simplest case.

The case of four tomatoes. Here it is.

And there is only one salad with the four tomatoes because there is

no room left for other ingredients so one salad.

Let's proceed to the next case.

The case of three tomatoes.

Now we have to pick the last ingredient.

It cannot be a tomato because we only have three tomatoes now.

And so it can be either bell pepper or lettuce.

The bell pepper, lettuce.

So there are two salads with three tomatoes.

Okay, next, case two tomatoes.

Now we have two ingredients left and they can be either two bell peppers,

one bell pepper and one lettuce and two lettuce.

Okay we have three salads in this case.

Now the case of one tomato.

Now we have three ingredients left.

It can either be- all of them can be bell pepper,

or two bell peppers and one lettuce,

one bell pepper and two lettuce, and three lettuce.

So there are four salads now.

In the last case. The case of zero tomato.

There are no tomatoes in our salad.

Then either all ingredients can be bell pepper,

there can be the three bell peppers and one lettuce.

There can be two lettuce and two bell peppers,

one bell pepper and three lettuce,

and all of them can be bell peppers.

So these were five cases.

So in the case of zero tomatoes there are five possible salads.

So we have considered all possible cases of the number of tomatoes.

And in total if the sun gets up,

we will see that these are 15 salads.

In total we have 15 salads.

So here is a list of all of our cells.

In the first column you can see one cell at four- four tomatoes.

Two possible salads for three tomatoes,

then three possible salads for two tomatoes,

then four options for one tomato,

and finally, five possible salads for zero tomatoes.

So here they are all on the picture.

There are 15 options in total and we are done.

Okay, the solution looks very structured.

For four tomatoes we have one salad,

for three tomatoes we have have two salads,

and so on each time the number of salads increases by one.

If we increase the size of a salad but keep the number of ingredients the same,

then the same structure holds.

So the solution will have the same structure.

But if we increase the number of ingredients,

then the picture becomes more complicated.

Yet the same strategy works and you can

still count the number of salads in the same recursive manner.