Problem 154
Exploring Pascal’s pyramid
A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.
Then, we calculate the number of paths leading from the apex to each position:
A path starts at the apex and progresses downwards to any of the three spheres directly below the current position.
Consequently, the number of paths to reach a certain position is the sum of the numbers immediately above it (depending on the position, there are up to three numbers above it).
The result is Pascal’s pyramid and the numbers at each level n are the coefficients of the trinomial expansion
(x + y + z)n.
How many coefficients in the expansion of (x + y + z)200000 are multiples of 1012?
探索帕斯卡四面体
我们用球构建一个三角形四面体，每一个球的下一层都由恰好三个球支撑。
然后，我们计算从顶端到每一个位置的路径数：
每一条路径从顶端出发，然后每次都走向下一层支撑当前位置的球的三个球之一。
最终，到达每个位置的路径数是它上方的球的路径数之和（根据位置的不同，每个位置的上方最多可能有三个球）。
最终的结果被称为帕斯卡四面体，这个四面体第n层的结果将是三项式(x + y + z)n的系数。
在三项式(x + y + z)200000的展开式中，有多少个系数是1012的倍数？