There is a box with 100 dice in it. These dice can be 4 sided, 8 sided, or 12 si

Question

There is a box with 100 dice in it.

These dice can be 4 sided, 8 sided, or 12 sided.

20 dice of the 100 are randomly selected and rolled.
The sides have corresponding numbers on them.

(4 sided will have sides labeled 1 through 4 and 8 sided dice will have sides labeled 1 through 8 and the same for 12 sided dice)

After the 20 dice are rolled, the sides add up to a sum of 92.

The amount of 8 sided dice is more than 4 sided. There are more 12 sided dice than 8 sided dice.
(the number of 4 sided dice < 8 sided dice < 12 sided dice)

how many of each dice are there?

Explanation / Answer

Let say there are number of 4 -sided , 8 - sided and 12- sided are x1 , x2 and x3 respectively.

x1 + x2 + x3 = 20

and x1 < x2 < x3

we can analyse that as all these values are integers

so minimum value x3  can take is 8 and maximum value x1 can take is  5

and as sum of all dices are equal to 92.

We can see that avergae value of dice = 92/20 = 4.6

so number of 8 sided dices must have values in between 6 and 7.

E(4 sided die) = 2.5

E(8 sided die) = 4.5

E(12 sided die) = 6.5

so possible values of combination of (x1,x2,x3) are (4,7,9) (5,7,8),(5,6,9)  

E(4,7,9) = 4 * 2.5 + 7 * 4.5 + 9 * 6.5 = 100

E(5,6,9) = 5 * 2.5 + 6 * 4.5 + 9 * 6.5 = 98

E(5,7,8) = 5 * 2.5 + 7 * 4.5 + 8 * 6.5 = 96

96 is the minimum expected value for any combination.

so we can say that

Number of 4 - sided die = 5

Number of 8 - side die = 7

Number of 12 - sided die = 8

(5,7,8)

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