If-1-x-n-C-0-C-1-x-C-2-x-2-C-3-x-3-C-n-x-n-then-prove-that-0-i-lt-j-n-i-n-C-i-j-n-C-j-n-2-2-r-0-n-1-n-C-r-

Question Number 23250 by Tinkutara last updated on 28/Oct/17

I f {(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3}

+ \dots + C_{n} x^{n}, t h e n p r o v e t h a t

\underset{0 ⩽ i < j ⩽ n}{Σ Σ} (\frac{i}{^{n} C_{i}} + \frac{j}{^{n} C_{j}}) = \frac{n^{2}}{2} (\underset{r = 0}{\sum^{n}} \frac{1}{^{n} C_{r}}) .