The coefficient of xk in 1-1-x-1-x2-1-xn- where 0-k-n is

Let S=1+(1+x)+(1+x)^2+⋯+(1+x)^n This is a G.P., so the sum is nbsp; ⇒ S=(1+x)^n+1 1(1+x) 1 ⇒ S=[(1+x)^n+1 1]x ⇒ S nbsp;= (∑_r = 0^n+1 ^n+1C_rx^r) 1x ∴ Coef ...

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Standard XII

Mathematics

Sum of Binomial Coefficients with Alternate Signs

The coefficie...

Question

The coefficient of $x^{k}$ in $1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ , where $0 \leq k \leq n$ is

^{n - 1} C_{k - 1}

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^{n} C_{k}

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^{n + 1} C_{k + 1}

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^{n + 1} C_{k}

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Solution

The correct option is C $^{n + 1} C_{k + 1}$
Let $S = 1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$
This is a G.P., so the sum is
$\Rightarrow S = \frac{(1 + x)^{n + 1} - 1}{(1 + x) - 1} \Rightarrow S = \frac{[(1 + x)^{n + 1} - 1]}{x} \Rightarrow S = \frac{(n + 1 \sum r = 0^{n + 1} C_{r} x^{r}) - 1}{x}$

$∴$ Coefficient of $x^{k}$ in $S =^{n + 1} C_{k + 1}$

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Sum of Binomial Coefficients with Alternate Signs

Standard XII Mathematics

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The coefficient of xk in 1+(1+x)+(1+x)2+⋯+(1+x)n, where 0≤k≤n is

The correct option is C n+1Ck+1Let S=1+(1+x)+(1+x)2+⋯+(1+x)n This is a G.P., so the sum is ⇒S=(1+x)n+1−1(1+x)−1⇒S=[(1+x)n+1−1]x⇒S=(n+1∑r=0 n+1Crxr)−1x ∴ Coefficient of xk in S= n+1Ck+1

The coefficient of $x^{k}$ in $1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ , where $0 \leq k \leq n$ is