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The coefficient of `x^(4)` in the expansion of `(1+x+x^(2)+x^(3))^(11)` is (where `({:(n),(r):})`=^(n)C_(r))`A. `({:(11),(4):})`B. `({:(11),(4):})+({:(11),(2):})`C. `({:(11),(4):})+({:(11),(2):})+({:(11),(4):}).({:(11),(2):})`D. `({:(11),(4):})=({:(11),(2):})+({:(11),(1):})({:(11),(2):})` |
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Answer» Correct Answer - D `1+x+x^(2)+x^(3)=(1+x)+x^(2)(1+x)=(1+x)(1+x^(2))` `(1+x+x^(2)+x^(3))^(11)=(1+x)^(11)(1+x^(2))^(11)` We want the coefficient of `x^(4)` `(1+^(11)C_(1).x+^(11)C_(2).x^(2)+^(11)C_(3).x^(3)+^(11)C_(4).x^(4)+…)` `xx(1xx^(11)C_(1).x^(2)+^(11)C_(2).x^(4)+....)` Collecting the terms which give `x^(4)` `1.^(11)C_(2).x^(4)+^(11)C_(2).x^(2).^(11)C_(1).x^(2)+^(11)C_(4)x^(4).1` `because^(11)C_(2)+^(11)C_(2)` .^(11)C_(1)+^(11)C_(4)=55+605+330=990` |
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