First slide

Combinations

Question

$The value of \sum_{r = 1}^{5} r \frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} =$

Moderate

A

$5 (n - 3)$
B

$5 (n - 2)$
C

5n
D

$5 (2 n - 9)$

Solution

$\begin{matrix} \frac{r \cdot C (n, r)}{C (n, r - 1)} = \frac{r \cdot n!}{(n - r)! r!} \cdot \frac{(r - 1)! (n - r + 1)!}{n!} \\ = n - r + 1 \\ \sum_{r = 1}^{5} \frac{r \cdot C (n, r)}{C (n, r - 1)} = n + (n - 1) + (n - 2) + (n - 3) + (n - 4) \\ = 5 n - 10 \end{matrix}$

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