The coefficient of x r 0 - r - n -1 in the expansion of x -3 n -1- x -3 n -2 x -2- x -3 n -3 x -22- x -2 n -1-A- n C r 3 n - r -2 n - r B- n C r -3 n 2 r -2 n 3 r -3-2 r -C- n C r 3 r -2 n - r D- n C n - r 3 r -2 n

The correct options are A ^nC_r(3^n r 2^n r) B ^nC_r[3^n2^r 2^n3^r/3^r2^r ]Let S=(x+3)^n 1+(x+3)^n 2(x+2)+(x+3)^n 3(x+2)^2+⋯ +(x+2)^n 1 The expression in Geomet ...

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Byju's Answer

Standard XII

Mathematics

Binomial Theorem for Any Index

The coefficie...

Question

The coefficient of $x^{r} (0 \leq r \leq (n - 1))$ in the expansion of $(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}$ .

^{n} C_{r} (3^{n - r} - 2^{n - r})

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^{n} C_{r} [\frac{3^{n} 2^{r} - 2^{n} 3^{r}}{3^{r} 2^{r}}]

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^{n} C_{r} (3^{r} - 2^{n - r})

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^{n} C_{n - r} (3^{r} - 2^{n})

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Solution

The correct options are
A $^{n} C_{r} (3^{n - r} - 2^{n - r})$
B $^{n} C_{r} [\frac{3^{n} 2^{r} - 2^{n} 3^{r}}{3^{r} 2^{r}}]$
Let $S = (x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}$
The expression in Geometric progression
$S = (x + 3)^{n - 1} (\frac{{(\frac{x + 2}{x + 3})}^{n} - 1}{\frac{x + 2}{x + 3} - 1}) = \frac{(x + 3)^{n - 1}}{(x + 3)^{n}} (\frac{(x + 2)^{n} - (x + 3)^{n}}{\frac{- 1}{(x + 3)}})$
$\Rightarrow S = - (x + 2)^{n} + (x + 3)^{n}$
Coefficient of $x^{r}$ in S is
$=^{n} C_{n - r} 3^{n - r} -^{n} C_{n - r} 2^{n - r}$
$=^{n} C_{r} (3^{n - r} - 2^{n - r})$
$=^{n} C_{r} (\frac{3^{n} 2^{r} - 2^{n} 3^{r}}{3^{r} 2^{r}})$

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Similar questions

Q. The coefficient of

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Q. Coefficient of

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If f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 + x^{n} & (1 - x)^{n} & 2 + x^{n} (2 + x)^{n} & (2 + x)^{n} & 1 (3 - x)^{n} & 1 & 3 + x \end{matrix} ∣ ∣ ∣ ∣ ∣

then the constant term in the expansion is

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The coefficient of xr(0≤r≤(n–1)) in the expansion of (x+3)n−1+(x+3)n−2(x+2)+(x+3)n−3(x+2)2+⋯+(x+2)n−1.

The coefficient of $x^{r} (0 \leq r \leq (n - 1))$ in the expansion of $(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}$ .