What is the probability that three 100-sided dice would exceed 222?
I assume each die has sides numbered from 1 to 100.
There are possible outcomes.
Visualize a cube graphed in the first octant of an xyz-system.
One vertex is at the origin; the opposite vertex is at (100, 100, 100).
Consider all the lattice points (those with integer coordinates)
. . from (1,1,1) to (100,100,100).
These represent the 1,000,000 possible outcomes.
The points whose coordinates have a sum exceeding 222
. . are "outside" the triangle with coordinates:
. . (22, 100, 100), (100, 22, 100), (100, 100, 22).
How many lattice points are contained in this tetrahedron?
The tetrahedron has 21 "levels".
Each level contains a triangular number of points.
We want the sum of the first 21 triangular numbers.
Fortunately, there is a formula for this: .