Number of elements in a flattened power set

Say $N$ is the number we are looking for. Say $n$ is the size of given set. Number of subsets of size r: $\binom{n}{r}$ . Total number of elements in power set given by subsets of size r $r\times\binom{n}{r}$ . Hence $N = \sum_{r=1}^n r \times \binom{n}{r}$ .

Say $B(x) = (1+x)^n$ .

$B(x) = \sum_{r=0}^n \binom{n}{r} x^r$

$B'(x) = \sum_{r=1}^n r\times\binom{n}{r} x^{r-1}$

$B'(1) = \sum_{r=1}^n r \times \binom{n}{r}$

Hence $N = B'(1)$

But also $B'(x) = n(1+x)^{n-1}$

and $B'(1) = n(2)^{n-1}$

Hence $N = n\times2^{n-1}$

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$