Intuitive proof of the formula ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$ I came across this formula in combination— ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$. Even though I know its rigorous mathematical proof, I want a logical and elegant proof of this.
For example, the famous formula of combination ${}_nC_r = {}_nC_{n-r}$ says selecting $r$ objects out of $n$ objects is same as rejecting $(n-r)$ objects.
So, I am looking for such kind of intuitive proof of the formula ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$, which I am unable to get. 
The thought of the wise man who said "writing a correct equation but not being able to interpret its result is the same as writing a grammatically correct sentence without knowing what it means!!!" is not helping me either!!
Note— ${}_nC_r$ means combination of $n$ objects taken $r$ at a time. 

Assume the $n+1$ objects are balls, with only one of them being red and the rest white.
A particular selection of $r$ balls has to either contain the red ball or not.
Those selections omitting the red ball are $^{n}C_r$. Those selections containing the red ball have choice in which $r-1$ white balls have to be chosen, and this is given by $^{n}C_{r-1}$.
