Q:

Verify the following identity by a committee-selection model

$${n \choose 0} + {n+1 \choose 1} + {n+2 \choose 2} +\cdots + {n + r \choose r} = {n+r+1 \choose r}$$
The "committee selection model" is a model used to view combinations as selecting a committee from a group of people. E.g., ${15 \choose 4}$ is the number of 4-person committees that can be formed from a group of 15 people. 
What I'm tasked with doing is a combinatorial proof using this model--showing that the left counts the same thing as the right using the committee model.

A:

On the RHS, you count the number of committees of $r$ people that can be chosen from a group on $n+r+1$ people.
Now select one particular person of the group, say John. Then you have two mutually exclusive cases:

John is not chosen in the committee, so the number of committees that ca be formed without John is ${n+r\choose r}$
John is chosen in the committee, so then you have to choose $r-1$ people out of a group of $n+r$. Now consider another person from the group, say Susie
2.1 John is chosen in the comitee, but Susie is not, which can be done in ${n+r-1\choose r-1}$ ways
2.2 Both John and Susie are chosen, in which case...

... and so on and so forth...
So if you label the people as $1,2,\ldots,n+r+1$, then the number of ways to choose $r$ people equals the sum of the ways to form the committee by mandatory choosing the labels $0,1,\ldots k-1$ and not to choose label $k$, for $k=1,2\ldots,r+1$

