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Combination

Prove that ...

Question

Prove that $n \sum r = 0 n_{C_{r}} 3^{r} = 4^{n}$

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Solution

$n \sum r = 0^{n} C_{r} 3^{r} = 4^{n}$
$(1 + x)^{n} =^{n} C_{r} +^{n} C_{r} x +^{n} C_{r} x^{2} +^{n} C_{r} x^{3} \dots .$
$x = 3$
$4^{n} =^{n} C_{r} +^{n} C_{r}^{3} +^{n} C_{r} x^{2}$
$4^{n} = n \sum r = 0^{n} C_{r} 3^{r}$ .

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