What is - r-0 n C n-r equal to-

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Sum of Coefficients of All Terms

What is ∑ r...

Question

What is $n \sum r = 0 C (n, r)$ equal to?

2^{n} - 1

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n

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n

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2^{n}

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Solution

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{(1 + x)}^{n} = \sum_{r = 0}^{n} C_{r} x^{r}

C_{0} + \frac{C_{1}}{2} + . . . . + \frac{C_{n}}{n + 1} = \frac{2^{n + 1} - 1}{n + 1}

(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{n},

\frac{C_{0}}{1 \cdot 2} 2^{2} + \frac{C_{1}}{2 \cdot 3} 2^{3} + \frac{C_{2}}{3 \cdot 4} 2^{4} + \dots + \frac{C_{n}}{(n + 1) (n + 2)} 2^{n + 2}

f (x) = x^{n}

f (1) + \frac{f^{'} (1)}{1} + \frac{f^{''} (1)}{2!} + \frac{f^{'''} (1)}{3!} + . . . + \frac{^{n} f (1)}{n!} = 2^{n}

{(1 + x)}^{n} = n \sum r = 0^{n} C_{r} x^{r}

n \sum r = 0^{n} C_{r} = 2^{n}

2 \leq r \leq n, (\frac{n}{r}) + 2 (\frac{n}{r - 1}) + (\frac{n}{r - 2})

n \sum r = 0 r^{2} . (^{n} C_{r})^{2}

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