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Sum of Coefficients of All Terms

For r=0,1, ...

Question

For $r = 0, 1, \dots$ , 10, let $A_{r}, B_{r}$ and $C_{r}$ denote, respectively, the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ Then $10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

A

B_{10} - C_{10}

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B

A_{10} (B_{10}^{2} - C_{10} A_{10})

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C

0

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D

C_{10} - B_{10}

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Solution

The correct option is D $C_{10} - B_{10}$
The given expansion can be written as $\sum_{r = 1}^{10} 10 C_{r} (20 C_{10} .20 C_{r} - 30 C_{10} .10 C_{r})$
$20 C_{10} \sum_{r = 1}^{10} 10 C_{r} .20 C_{r} - 30 C_{10} \sum_{r = 1}^{10} {(10 C_{r})}_{r}^{2}$
$\Rightarrow 20 C_{10} [30 C_{20} - 1] - 30 C_{10} [20 C_{10} - 1]$
$\Rightarrow 30 C_{10} - 20 C_{10}$
$\Rightarrow C_{10} - B_{10}$

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Similar questions

r = 0, 1, . . ., 10

A_{r}, B_{r},

C_{r}

denote, respctively, the coefficient of

x^{r}

in the expansions of

(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

A_{r}, B_{r}, C_{r}

denotes the coefficients of

x^{r}

in the expansion of

(1 + x)^{10}, (x + 1)^{20}, (1 + x)^{30}

respectively, then the value of

10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, . . ., 10

A_{r}, B_{r}

C_{r}

denote, respectively, the coefficient of

x^{r}

in the expansions of

(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, . . . ., 10

A_{r}, B_{r}

C_{r}

denote, respectively, the coefficient of

x^{r}

in the expansions of

{(1 + x)}^{10}, {(1 + x)}^{20}

{(1 + x)}^{30}

\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, 2, 3, . . ., 10

A_{r}, B_{r}, C_{r}

denote respectively the coefficient of

x^{r}

in the expansions of

(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

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