Here's the problem from the book
I thought it'd be helpful to have an example
We can represent this like a graph with n nodes (I'm keeping 15 to match with our previous example, 3 + 5 + 7 = 15)
In a graph, "choose 2" means just means count all of the edges in the fully connected graph. Because each edge is "choosing" two nodes to connect. In this case, 15 choose 2 = 105
Now lets break that full graph down into three sub-graphs of sizes k_1, k_2, ..., k_j totalling to n
For each of these, k_i choose 2 is just asking for the number of edges in the complete subgraph
This is necessarily less than the number of edges in the complete graph, because all of the edges between the subgraphs are missing