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Binomial Coefficients

For r=0, 1,...

Question

For $r = 0, 1, 2, 3, . . ., 10$ , let $A_{r}, B_{r}, C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ 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 C $C_{10} - B_{10}$
$A_{r}$ =coefficient of $x^{r}$ in ${(1 + x)}^{10} =^{10} C_{r}$
$B_{r}$ =coefficient of $x^{r}$ in ${(1 + x)}^{20} =^{20} C_{r}$
$C_{r}$ =coefficient of $x^{r}$ in ${(1 + x)}^{30} =^{30} C_{r}$
$∴ \sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$
$= \sum_{r = 1}^{10} A_{r} B_{10} B_{r} - \sum_{r = 1}^{10} A_{r} C_{10} A_{r}$
$= \sum_{r = 1}^{10}^{10} {C_{r}}^{20} {C_{r}}^{20} C_{r} - \sum_{r = 1}^{10}^{10} {C_{r}}^{30} {C_{10}}^{10} C_{r}$
$= \sum_{r = 1}^{10}^{10} {C_{10 - r}}^{20} {C_{10}}^{20} C_{r} - \sum_{r = 1}^{10}^{10} {C_{10 - r}}^{30} {C_{10}}^{10} C_{r}$
$=^{20} C_{r} \sum_{r = 1}^{10}^{10} {C_{10 - r}}^{20} {C_{r} -}^{30} C_{10} \sum_{r = 1}^{10}^{10} {C_{10 - r}}^{10} C_{r}$
$=^{20} C_{10} (^{30} C_{10} - 1) -^{30} C_{10} (^{20} C_{10} - 1)$
$=^{30} C_{10} -^{20} C_{10} = C_{10} - B_{10}$

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Similar questions

r = 0, 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, \dots

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})

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})

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