Counting Dice Outcomes
(Multinomials)
The formula...
One multinomial to assign trials to piles
One multinomial to assign faces (ranks) to pile sizes
n = number of trials (rolls)
F = number of faces (sides) on the die
Problem 1
If we roll a 6-sided die 5 times, in how many ways can we get 3 of a kind (3OAK) and 2 of a kind (2OAK)?
(Click the Down Arrow to see the solution)
3OAK 2OAK 0OAK 0OAK 0OAK 0OAK
Intrapile 3 2 0 0 0 0
Interpile 1 1 4 of size 3 of size 2 of size 0
(Click Right Arrow for the next problem)
There are F=6 piles--one for each side of the die.
F=6 piles →
↖ 1 pile of size 3, 1 pile of size 2, 4 piles of size 0
Problem 2
Using an 8-sided die, in how may ways can we get 3OAK, 3OAK, and 1OAK?
(Click Down Arrow for the solution)
(Click Right Arrow for the next problem)
F=8 piles →
↖ 2 piles of size 3, 1 of size 1, 5 of size 0
Problem 3
Using a die with 5 sides, in how many ways can we get 2OAK, 2OAK, and 2OAK?
(Click Down Arrow for the solution)
F=5 piles →
↖ 3 piles of size 2, 2 piles of size 0
Fin
Dice Outcomes Practice Problems
By smilinjoe
Dice Outcomes Practice Problems
- 667