How Many Unique Ways Can 4 Dice Be Combined?

[nille40]

Hi all!

In how many unique ways can 4 dices be combined? Note that the order amongst the dices is not relevant, so 1-2-3-4 = 4-3-2-1.

My idea is that you select the values, one by one. You can select the first value in 6 ways, the second in 6 ways, the third in 6 ways and the fourth in 6 ways. This yield $$6^4$$ combinations. The order was irrelevant, so the answer should then be $$\frac{6^4}{4!}$$.

This is obviously wrong... I'm trying to figure out how to think to solve a problem like this.

The answer is

$${6+4-1} \choose {4}$$

Which basically means "select 4 of the 6, and put each value back when you've selected it". I don't get this...

Would really appreciate some guidance!
Nille

 

[nille40]

Ok, I have an idea.

Lets say we have dices in a line. The first dice has the value 1, the second 2, the third 3...the sixth 6. This yields the equation

$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 4$$

So the solution $$x_1 = 2, x_3 = 1, x_4 = 1$$ means that two dices has the value 1, one has 3 and one has 4.

This equation has the solution
$${{4 + 6 - 1} \choose {4}} = {{4+6-1} \choose {5}}$$

Can this be solved in some other way?
Nille