First slide

Combinations

Question

The value of $\sum_{r = 1}^{n} \frac{{n_{P}}_{r}}{r!}$ , is

Moderate

A

$2^{n}$
B

$2^{n} - 1$
C

$2^{h} - 1$
D

$2^{n} + 1$

Solution

We have
$\begin{array}{l} ^{n} P_{r} =^{n} C_{r} r! \\ \Rightarrow \frac{^{n} P_{r}}{r!} =^{n} C_{r} \\ \Rightarrow \sum_{r = 1}^{n} \frac{^{n} P_{r}}{r!} = \sum_{r - 1}^{n}^{n} C_{r} =^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n} = 2^{n} - 1 \end{array}$

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