The value of $2000_{C_{2}} + 2000_{C_{5}} + 2000_{C_{8}} + . . . + 2000_{C_{2000}} = ?$

\frac{2^{1999} - 1}{3}

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\frac{2^{1999} + 1}{3}

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\frac{2^{2000} + 1}{3}

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\frac{2^{2000} - 1}{3}

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Solution

The correct option is D $\frac{2^{2000} - 1}{3}$
$(1 + x)^{n} = n_{C_{0}} + n_{C_{1}} x + n_{C_{2}} x^{2} + . . . + n_{C_{n}} x^{n}$
Put x = 1, $w$ , $w^{2}$ and add
$\Rightarrow C_{2} + C_{5} + C_{8} + . . . . = \frac{1}{3} {2^{n} + (- 1)^{n} (w^{2 n + 1} + w^{n + 2})}$
$= \frac{1}{3} {2^{2000} + (- 1)^{2000} (w^{4001} + w^{2002})}$
$= \frac{2^{2000} - 1}{3}$

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Similar questions

The value of $2000_{C_{2}} + 2000_{C_{5}} + 2000_{C_{8}} + . . . + 2000_{C_{2000}} = ?$

A. $\frac{2^{1999} + 1}{3}$

B. $\frac{2^{1999} - 1}{3}$

C. $\frac{2^{2000} + 1}{3}$

D. $\frac{2^{2000} - 1}{3}$

\int_{- 2}^{2} (a x^{3} + b x + c)

A B C

y

x

z

y

x

The value of $\int_{- 2}^{2} (a x^{3} + b x + c)$ depends on [MNR 1988; UPSEAT 2000]

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