# A boy wishes to buy exactly six marbles.There are 4 different colours of marbles available.In how many ways can he buy six marbles?

$\left(\begin{matrix}9 \\ 3\end{matrix}\right) = 84$

#### Explanation:

Let's think of this in this way - if we think of the problem as involving two different qualities - that of being a marble, and that of being the colour, we can place the marbles (identical) into different colour baskets (unique). There are 4 colour baskets into which we'll place 6 marbles.

So let's lay out the marbles (I'll lay out the marbles with the letter m):

mmmmmm

And I'll use the character | to indicate the walls of the baskets. We can move the walls to indicate which basket gets how many marbles (we only need 3 walls to differentiate the baskets). So a couple of ways we get the baskets aligned is:

mmm|m|m|m - which has 3 of one colour and 1 each of 3 colours

and |||mmmmmm - which has 6 marbles of 1 colour.

For those familiar with the technique, this is called stars and bars (usually the m is placed with a star or *).

We have 6 marbles to place, as well as 3 bars, so that's 9 items in the population. We're now choosing bars, and so we choose 3:

$\left(\begin{matrix}9 \\ 3\end{matrix}\right) = 84$

I find the explanation on this site to be very helpful in guiding my explanations:

https://brilliant.org/wiki/identical-objects-into-distinct-bins/