1. ## binomial coefficients

if $C_r =\frac {n!}{(n-r)!(r)!}$

prove

$\frac{C_1}{C_0} + 2\frac{C_2}{C_1} +3\frac{C_3}{C_2} + ... +n\frac{C_n}{C_(n-1)} = \frac{n(n+1)}{2}$

2. use $\frac{C_{r}}{C_{r-1}} = \frac{n[n-(r-1)](r-1)!}{r!}$

3. Also try to link in the fact that

$1+2+3+\cdots+(n-1)+n=S$ $[1]$

where S is the sum of terms, then making the order of the LHS opposite

$n+(n-1)+\cdots+3+2+1 = S$ $[2]$

adding $[1]+[2]$ term for term gives

$(n+1)+(n+1)+(n+1)+\cdots+(n+1)+(n+1) = 2S$

$n(n+1) = 2S$

$\frac{n(n+1)}{2} = S$