If the constant term, in binomial expansion of ${\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}$ is 180, then r is equal to __________________.
Answer (integer)
8
Solution
${\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}$<br><br>General term $= {}^{10}{C_R}{(2{x^2})^{10 - R}}{x^{ - 2R}}$<br><br>$\Rightarrow {2^{10 - R}}{}^{10}{C_R} = 180$ ....... (1)<br><br>& (10 $-$ R)r $-$ 2R = 0<br><br>$r = {{2R} \over {10 - R}}$<br><br>$r = {{2(R - 10)} \over {10 - R}} + {{20} \over {10 - R}}$<br><br>$\Rightarrow r = - 2 + {{20} \over {10 - R}}$ ....... (2)<br><br>R = 8 or 5 reject equation (1) not satisfied<br><br>At R = 8<br><br>$\Rightarrow {2^{10 - R}}\times{}^{10}{C_R} = 180 \Rightarrow r = 8$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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