First slide

Permutations

Question

If $a_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}},$ then value of $\sum_{r = 0}^{n} \frac{n - 2 r}{^{n} C_{r}}$ is

Moderate

A

$\frac{n}{2} a_{n}$
B

$\frac{1}{4} a_{n}$
C

$n a_{n}$
D

0

Solution

$\begin{array}{l} \sum_{r = 0}^{n} \frac{n - 2 r}{^{n} C_{r}} = \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{r}} - \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} \\ = \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{n - r}} - \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} = 0 \end{array}$

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