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Sum of Coefficients of All Terms

For r = 0, ⋯ ...

Question

For $r = 0, \dots, 10, I f A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}, a n d (1 + x)^{30}$ . Then the value of $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is

A

B_{10} - C_{10}

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B

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

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C

\N

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D

C_{10} - B_{10}

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Solution

The correct option is D $C_{10} - B_{10}$
$A_{r}$ = Coefficient of $x^{r} i n (1 + x)^{12} =^{10} C_{r}$
$B_{r}$ = Coefficient $o f x^{r} i n (1 + x)^{20} =^{20} C_{r}$
$C_{r}$ = Coefficient of $x^{r} i n (1 + x)^{30} =^{30} C_{r}$
$∴ \sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r}) = \sum_{r = 1}^{10} A_{r} B_{10} B_{r} - \sum_{r = 1}^{10} A_{r} C_{10} A_{r}$
= $\sum_{r = 1}^{10}^{10} C_{r}^{20} C_{10}^{20} C_{r} - \sum_{r = 1}^{10}^{10} C_{r}^{30} C_{10}^{10} C_{r}$
$= \sum_{r = 1}^{10}^{10} C_{10 - r}^{20} C_{10}^{20} C_{r} - \sum_{r = 1}^{10}^{10} C_{10 - r}^{30} C_{10}^{10} C_{r}$
$=^{20} C_{10} \sum_{r = 1}^{10}^{10} C_{10 - r} .^{20} C_{r} -^{30} C_{10} \sum_{r = 1}^{10}^{10} C_{10 - r}^{10} C_{r}$
$=^{20} C_{10} (^{30} C_{10} - 1) -^{30} C_{10} (^{20} C_{10} - 1) =^{30} C_{10} -^{20} C_{10} = C_{10} - B_{10}$

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Similar questions

r = 0, \dots, 10, I f A_{r}, B_{r}

C_{r}

denote respectively the coefficient of

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in the expansions of

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

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