I've tried to answer the following problem:
In my mind, I thought about a very complicated solution: First counting $i=j\geq k$ divided in various cases and then taking appropriate permutations $k=j\geq i$, etc. But the solution in the book is:
It is so simple that I don't know what is happening in there, I don't know how he came up with $\sum_{i=1}^{75} i$?
I am also confused about the given conditions: How did he came up with them and how is he actually using it?
First, let's obtain the conditions.
Clearly $i,j,k\le 75$. $$k=151-i-j\le 75$$ Then $i+j\ge 76$ and therefore $76-i \le j \le 75$.
The author then looks at each possibility for $i$. For each one, the number of possibilities for $j$ is $$75-(76-i)+1=i.$$
The answer is therefore the sum $$1+2+...+75=2850.$$