Dice game

Probability Level 4

20 identical dice are thrown. What is the total number of different outcomes?


The answer is 53130.

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Vijay Bhava by Vijay Bhava , 21, India

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2 solutions

Relevant wiki: Integer Equations - Stars and Bars

Let n k n_{k} represent the number of dice that show the value k k on their upward face. Then we are wanting to find the number of non-negative integer solutions to the equation n 1 + n 2 + n 3 + n 4 + n 5 + n 6 = 20 n_{1} + n_{2} + n_{3} + n_{4} + n_{5} + n_{6} = 20 .

This is a stars and bars calculation with solution ( 20 + 6 − 1 20 ) = ( 25 20 ) = 53130 \dbinom{20 + 6 - 1}{20} = \dbinom{25}{20} = \boxed{53130} .

4 Helpful 0 Interesting 0 Brilliant 0 Confused

yes sir exactly! use of stars and bars is the most efficient way.

Vijay Bhava - 4 years, 5 months ago

Since the dice are indistinguishable, we can use the formula for combination with repitition,

(n+5)C5 = 25C5 = 53130

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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