How Many Unique Ways Can 4 Dice Be Combined?

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The discussion centers on calculating the unique combinations of values when rolling four dice, where the order of the dice does not matter. The initial approach incorrectly applies the formula for permutations instead of combinations. The correct method involves using the "stars and bars" theorem, represented mathematically as {6+4-1} choose {4}, which accounts for selecting values with repetition. A further explanation illustrates how to set up an equation to represent the distribution of values across the dice. The conversation seeks clarification on alternative methods for solving this combinatorial problem.
nille40
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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
 
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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
 
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