Question

If (1 + x)ⁿ = $\sum_{r = 0}^{n}{nC_{r}x^{r}}$, then $\sum_{r = m}^{n}{rC_{m}}$ is equal to

• A. ^{n + 1}C_m
• B. ^{n + 1}C_{m + 1}
• C. ^{n + 2}C_{m + 1}
• D. None of these

Answer 1

Answer: (B)

^{n + 1}C_{m + 1}

Answer 2

C_m is coefficient of x^m in (1 + x)^r

⇒ $\sum_{r = m}^{n}{rC_{m}}$is coefficient of x^m in

(1 + x)^m + (1 + x)^{m + 1} + …… + (1 + x)ⁿ

i.e. coefficient of x^m in $\frac{(1 + x)^{m}\left\{ (1 + x)^{n + 1 - m} - 1 \right\}}{x}$

or coefficient of x^{m + 1} in (1 + x)^{n + 1} = ^{n + 1}C_{m + 1}