- r -1 n - k -0 r -1 n C r r C k 2 k is equal toA- 4 n -3 n -2B- 4 n -3 nC- 4 n -3 n -1D- 4 n -3 n -1

The correct option is B 4^n 3^n ∑_r=1^n ^nC_r ( ∑_k=0^n ^r C_k 2^k 2^n ) ∑_r=1^n ^nC_r (3^n 2^n)=4^n 3^n ...

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Binomial Coefficients

∑ r =1 n ∑ k ...

Question

$\sum_{r = 1}^{n} (\sum_{k = 0}^{r - 1}^{n} C_{r}^{r} C_{k} 2^{k})$ is equal to

$4^{n} - 3^{n}$

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$4^{n} - 3^{n} + 1$

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$4^{n} - 3^{n} - 1$

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$4^{n} - 3^{n} + 2$

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Solution

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lim n \to \infty \frac{- 3 n + (- 1)^{n}}{4 n - (- 1)^{n}}

\frac{1}{9} \times 3^{n} = 3^{n - 2}

64 \times 4^{n - 2} = 4^{n}

1. (2 - w) (2 - w^{2}) + 2. (3 - w) (3 - w^{2}) + . . . . . . . . . . + (n - 1) (n - w) (n - w^{2}),

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