The coefficient of xr 0- r- n-1 in the expression x-2 n-1- x-2 n-2- x-1 - x-2 n-3- x-1 2- - x-1 n-1 is

The correct option is B ^nC_r ( 2^n r 1 ) Given expression is a G.P. #160; Therefore its sum is ( x+2 )^n 1{ ( x+1/x+2 )^n 1/ ( x+1/x+2 ) 1}= ( x+ ...

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Standard XII

Mathematics

Binomial Coefficients

The coefficie...

Question

The coefficient of $x^{r} (0 \leq r \leq n - 1)$ in the expression ${(x + 2)}^{n - 1} + {(x + 2)}^{n - 2} \cdot (x + 1) + {(x + 2)}^{n - 3} \cdot {(x + 1)}^{2} + \cdot \cdot \cdot + {(x + 1)}^{n - 1}$ is

^{n} C_{r} (2^{r} - 1)

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^{n} C_{r} (2^{n - r} - 1)

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^{n} C_{r} (2^{r} + 1)

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^{n} C_{r} (2^{n - r} + 1)

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Solution

The correct option is B $^{n} C_{r} (2^{n - r} - 1)$
Given expression is a G.P. Therefore its sum is
${(x + 2)}^{n - 1} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \frac{{(\frac{x + 1}{x + 2})}^{n} - 1}{(\frac{x + 1}{x + 2}) - 1} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = {(x + 2)}^{n} - {(x + 1)}^{n}$
Now coefficient of $x^{r} =^{n} C_{r} 2^{n - r} -^{n} C_{r} =^{n} C_{r} (2^{n - r} - 1)$

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The coefficient of xr(0≤r≤n−1) in the expression (x+2)n−1+(x+2)n−2⋅(x+1)+(x+2)n−3⋅(x+1)2+⋅⋅⋅+(x+1)n−1 is

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The coefficient of $x^{r} (0 \leq r \leq n - 1)$ in the expression ${(x + 2)}^{n - 1} + {(x + 2)}^{n - 2} \cdot (x + 1) + {(x + 2)}^{n - 3} \cdot {(x + 1)}^{2} + \cdot \cdot \cdot + {(x + 1)}^{n - 1}$ is