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Sum of Coefficients of All Terms

The general t...

Question

The general term in the expansion of $(1 + 3 x - 2 x^{3})$ is

A

\frac{10!}{r_{1}! r_{2}! r_{3}!} (- 1)^{r_{3}} 3^{r_{2}} 2^{r_{3}} x^{r_{2} + 3 r_{3}}

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B

\frac{10!}{r_{1}! r_{2}! r_{3}!} (- 1)^{r_{2}} 3^{r_{3}} 2^{r_{3}} x^{r_{2} + 3 r_{3}}

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C

\sum r_{1} + r_{2} + r_{3} = 10

\frac{10!}{r_{1}! r_{2}! r_{3}!} 3^{r_{2}} 2^{r_{3}} {(- 1)}^{r_{3}} x^{r_{2} + 3 r_{3}}

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D

None of these

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Solution

The correct option is C $\sum r_{1} + r_{2} + r_{3} = 10$ $\frac{10!}{r_{1}! r_{2}! r_{3}!} 3^{r_{2}} 2^{r_{3}} {(- 1)}^{r_{3}} x^{r_{2} + 3 r_{3}}$
$(x_{1} + x_{2} + \dots + x_{k})^{n} = \sum r_{1} + r_{2} + r_{3} + \dots + r_{k} = n \frac{n!}{r_{1}! r_{2}! r_{3}! . . r_{k}!} x_{1}^{r_{1}} x_{2}^{r_{2}} . . . x_{n}^{r_{n}}$
$(1 + 3 x - 2 x^{3})^{10} = \sum r_{1} + r_{2} + r_{3} = 10 \frac{10!}{r_{1}! r_{2}! r_{3}!} (1)^{r_{1}} (3 x)^{r_{2}} (- 2 x^{3})^{r_{3}}$
General term is $\frac{10!}{r_{1}! r_{2}! r_{3}!} 3^{r_{2}} 2^{r_{3}} (- 1)^{r_{3}} x^{r_{2} + 3 r_{3}}$

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Similar questions

\frac{R_{1}}{r_{1}} + \frac{R_{2}}{r_{2}} + \frac{R_{3}}{r_{3}} =

Q. Assertion :In a

Δ

r_{1} + r_{2} + r_{3} - r = 4 R

r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = Δ^{2}

Q. Assertion :In a

△ A B C

a < b < c

r

is inradius and

r_{1}, r_{2}, r_{3}

are the axradii opposite to angle

A, B, C

respectively, then

r < r_{1} < r_{2} < r_{3}

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{r_{1} r_{2} r_{3}}{r}

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}

is equal to
(where

r_{1}, r_{2}

r_{3}

are the exradii and

2 s

is the perimeter.)

if $(1 + x + \frac{1}{x})^{4} = \sum_{r_{1} + r_{2} + r_{3}} = 4 \frac{4!}{r_{1}! \times r_{2}! \times r_{3}!} \times (1)^{r_{1}} (x)^{r_{2}} (\frac{1}{x})^{r_{3}}$ how many values can $r_{1}$ take so that the terms in the expansion will be independent of x.

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