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Sum of Coefficients of All Terms

If Ar,Br,Cr d...

Question

If $A_{r}, B_{r}, C_{r}$ denotes the coefficients of $x^{r}$ in the expansion of $(1 + x)^{10}, (x + 1)^{20}, (1 + x)^{30}$ respectively, then the value of $10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})$ 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

0

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D

C_{10} - B_{10}

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Solution

The correct option is D $C_{10} - B_{10}$
Given series
$10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r}) = B_{10} 10 \sum r = 1 A_{r} B_{r} - C_{10} 10 \sum r = 1 (A_{r})^{2}$

Now,
$(1 + x)^{10} = A_{0} + A_{1} x + A_{2} x^{2} + \dots + A_{10} x^{10}$
$(x + 1)^{20} = B_{0} x^{20} + B_{1} x^{19} + B_{2} x^{18} + \dots + B_{20}$

Now, coefficient of $x^{20}$ in $(1 + x)^{10} (x + 1)^{20}$ is $A_{0} B_{0} + A_{1} B_{1} + A_{2} B_{2} + \dots + A_{10} B_{10}$

As $(1 + x)^{10} (x + 1)^{20} = (1 + x)^{30}$
So, coefficient of $x^{20}$ is $C_{20}$

$\Rightarrow 10 \sum r = 0 A_{r} B_{r} = C_{20} = C_{10} ∴ 10 \sum r = 1 A_{r} B_{r} = C_{10} - 1 \dots (1)$

We know that
$(^{n} C_{0})^{2} + (^{n} C_{1})^{2} + \dots + (^{n} C_{n})^{2} =^{2 n} C_{n} \Rightarrow A_{0}^{2} + A_{1}^{2} + A_{2}^{2} + \dots + A_{10}^{2} =^{20} C_{10} ∴ 10 \sum r = 1 A_{r}^{2} = B_{10} - 1 \dots (2)$

Therefore,
$10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r}) = B_{10} 10 \sum r = 1 A_{r} B_{r} - C_{10} 10 \sum r = 1 A_{r}^{2}$
Using equation $(1)$ and $(2)$ , we get
$= B_{10} (C_{10} - 1) - C_{10} (B_{10} - 1) = C_{10} - B_{10}$

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Similar questions

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})

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

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

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