I'm studying for my final for Statistics, and I want to understand literally every problem in my textbook (at least in the first 7 chapters).  One of the problems asks to show that ${n}\choose {k}$ $=$${n-1}\choose {k}$ $+$ ${n-1}\choose {k-1}$  I have the solution manual, but the steps it gives appears to skip a few important steps or leave out important clarifications.  Can someone please guide me step by step through this proof?
I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas?
If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.  Just making sure I am on the right track...  Do I expand each piece out and solve algebraically?