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Ratio and Proportion

For r=0,1,2...

Question

For $r = 0, 1, 2,, . . . .10$ 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 $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ 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 $C_{10} - B_{10}$
$A_{r} =$ Coefficient of $x^{r}$ in $(1 + r)^{10} = 10 C_{r}$
$B_{r} =$ Coefficient of $x^{r}$ in $(1 + r)^{20} = 20 C_{r}$
$C_{r} =$ Coefficient of $x^{r}$ in $(1 + r)^{30} = 30 C_{r}$
$10 \sum r = 1 A r (B_{10} B_{r} - C_{10} A_{r}) = 10 \sum r = 1 A_{r} B_{10} B_{r} - 10 \sum r = 1 A_{r} C_{10} A_{r}$
$= 10 \sum r = 1_^{10} C_{r}_^{20} C_{10}_^{20} C_{r} - 10 \sum r = 1_^{10} C_{r}_^{30} C_{10}_^{10} C_{r}$
$= 10 \sum r = 1_^{10} C_{r}_^{20} C_{10}_^{20} C_{r} - 10 \sum r = 1_^{10} C_{r}_^{30} C_{10}_^{10} C_{r}$
$=_^{20} C_{10} 10 \sum r = 1_^{10} C_{10 - r}_^{20} C_{r} -_^{30} C_{10} 10 \sum r = 1_^{10} C_{10 - r}_^{10} C_{r}$
$=^{20} C_{10} (^{30} C_{10} - 1) -^{30} C_{10} (^{20} C_{10} - 1)$
$=^{30} C_{10} -^{20} C_{10} = C_{10} - B_{10}$
$C_{10} - B_{10}$

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Similar questions

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, . . ., 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})

Q. For r=0,1,2,........10, let

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, 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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