While finding sum of binomial coefficients,why or how can we change index of Σ?

Question

Like here, how and why did we change it? When can we change this, when can’t we change the index cause I know it matters? I have seen many examples now and have no source of information regarding this

Answer 1

If I have understood it correctly, the confusion you are having is how to write $$\sum_{r = 0}^{n}r \,^{n}C_{r} = n \sum_{r = 1}^{n}\,^{n-1}C_{r-1}.$$

The proof is here:

\begin{align} \sum_{r = 0}^{n}r \, ^{n}C_{r} & = \underbrace{0 \cdot \,^{n}C_{0}}_{=\, 0} + 1 \cdot \,^{n}C_{1} + \cdots + n \cdot \,^{n}C_{n} \\ & = 1 \cdot \,^{n}C_{1} + \cdots + n \cdot \,^{n}C_{n} = \sum_{r = 1}^{n}r\,^{n}C_{r} \\ & = \sum_{r = 1}^{n} \left[r \times \dfrac{n!}{(n-r)! \, r!} \right] = \sum_{r = 1}^{n} \left[r \times \dfrac{n \times (n-1)!}{(n-r)! \times \, r \times (r-1)!} \right]\\ & = n \left[ \sum_{r = 1}^{n}\dfrac{(n-1)!}{((n-1) - (r-1))!\,(r-1)!}\right] \\ & = n \sum_{r = 1}^{n} \, ^{n-1}C_{r-1} \end{align}