InterviewSolution
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InterviewSolution
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The term independent of x in the expansion of `((1)/(x^(2)) + (1)/(x) +1 + x + x^(2))^(5)`, isA. 381B. 441C. 439D. 359 |
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Answer» Correct Answer - a We have, `((1)/(x^(2)) + (1)/(x) + 1+ x + x^(2))^(5) = {(1)/(x^(2))((1 -x^(5))/(1 - x))}^(5) = (1)/(x^(10)) (1 - x^(5))^(5)(1 - x)^(-5)` Clearly, the term independent of x in ` ((1)/(x^(2)) + (1)/(x) + 1 + x + x^(2))^(5)` is equal to the coefficient of `x^(10)` in `(1 - x ^(5))^(5) (1 - x)^(-5)` . Now, `(1 - x^(5))^(5) (1 - x )^(-5) = (""^(5)C_(0) - ""^(5)C_(1)x^(5)+""^(5)C_(2)x^10-...)(sum_(r=0)^(infty)""^(4 +r)C_(r)x^(r))` `- (""^(5)C_(0) - ""^(5)C_(1)x^(5)+""^(5)C_(2)x^10-...)(sum_(r=0)^(infty)""^(4 +r)C_(4)x^(r))` `therefore ` Coefficient of `x^(10)` in `(1 - x ^(5))^(5) (1 - x)^(-5)` `=""^(5)C_(0) xx ""^(14)C_(4)-""^(5)C_(1)xx""^(9)C_(4)+""^(5)C_(2)xx""^(4)C_(4)= 381` |
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