How Can Ivan Achieve a Total Score of 5 with Three Dice?

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To achieve a total score of 5 with three dice, one must list all combinations of the dice that add up to this total. The valid combinations include (1, 1, 3), (1, 2, 2), and permutations thereof. Calculating the total outcomes for three dice reveals there are 216 possible combinations (6 sides per die). The probability of rolling a total score of 5 is thus calculated as the number of successful outcomes (6) divided by the total outcomes (216), resulting in a probability of 1/36. Understanding this process is essential for calculating probabilities with multiple dice.
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Ivan throws three fair dice.

(i) List all the possible scores on the three dice which give a total score of 5, and hence show that
the probability of Ivan obtaining a total score of 5 is 1/36


MY PROBLEM HERE IS THAT I CANNOT LIST THE PROBABILITY OF POSSIBLE SCORE FOR THREE DICE, IN FACT FOR THE TWO DICE I KNOW THAT I HAVE TO CONTRUCT A TABLE BUT FOR THE 3 DICE I AM STUCK




HELPPPP THANKS IN ADVANCE
 
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Start by making a table with three columns

Die 1 Die 2 Die 3

then fill in the rows with all the ways to have the sides occur so that the sum is 5. You'll also need to calculate the total number of outcomes (number of ways three dice can come up: the number for 2 dice is 36 - you need the corresponding number for three dice)
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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