jee-advanced 2010 Q20

jee-advanced · India · paper2 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients

For $\mathrm { r } = 0,1 , \ldots , 10$, let $\mathrm { A } _ { \mathrm { r } } , \mathrm { B } _ { \mathrm { r } }$ and $\mathrm { C } _ { \mathrm { r } }$ denote, respectively, the coefficient of $\mathrm { x } ^ { \mathrm { r } }$ in the expansions of $( 1 + \mathrm { x } ) ^ { 10 } , ( 1 + \mathrm { x } ) ^ { 20 }$ and $( 1 + \mathrm { x } ) ^ { 30 }$. Then
$$\sum _ { r = 1 } ^ { 10 } A _ { r } \left( B _ { 10 } B _ { r } - C _ { 10 } A _ { r } \right)$$
is equal to
A) $\mathrm { B } _ { 10 } - \mathrm { C } _ { 10 }$
B) $\mathrm { A } _ { 10 } \left( \mathrm {~B} _ { 10 } ^ { 2 } - \mathrm { C } _ { 10 } \mathrm {~A} _ { 10 } \right)$
C) O
D) $\mathrm { C } _ { 10 } - \mathrm { B } _ { 10 }$

For $\mathrm { r } = 0,1 , \ldots , 10$, let $\mathrm { A } _ { \mathrm { r } } , \mathrm { B } _ { \mathrm { r } }$ and $\mathrm { C } _ { \mathrm { r } }$ denote, respectively, the coefficient of $\mathrm { x } ^ { \mathrm { r } }$ in the expansions of $( 1 + \mathrm { x } ) ^ { 10 } , ( 1 + \mathrm { x } ) ^ { 20 }$ and $( 1 + \mathrm { x } ) ^ { 30 }$. Then

$$\sum _ { r = 1 } ^ { 10 } A _ { r } \left( B _ { 10 } B _ { r } - C _ { 10 } A _ { r } \right)$$

is equal to\\
A) $\mathrm { B } _ { 10 } - \mathrm { C } _ { 10 }$\\
B) $\mathrm { A } _ { 10 } \left( \mathrm {~B} _ { 10 } ^ { 2 } - \mathrm { C } _ { 10 } \mathrm {~A} _ { 10 } \right)$\\
C) O\\
D) $\mathrm { C } _ { 10 } - \mathrm { B } _ { 10 }$

Paper Questions

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