Simple combinatorics

A simple explanation for the following result
^NC_r=^N-1C_r+^N-1C_r-1

Suppose you have 100 people and you need to choose 10 people. How many ways can you do this? This is equal to ^NC_r

You can break this into two parts.

1. How many ways can you choose 10 people from 99 and excluding yourself from the 10? This is the same as ^N-1C_r

2. How many ways can you choose 9 people from 99 and keeping yourself as the 10th? This is the same as
  ^N-1C_r-1

It is clear that summing these two parts gives us the number of ways to choose 10 people from 100.
I heard this in a machine learning course offered by Caltech. Machine Learning lecture