If the constant term, in binomial expansion of ( 2x^r + 1 / x^2 )^10 is 180, then r is equal to __________________.

( 2x^r + 1 / x^2 )^10 General term = ^10C_R(2x^2)^10 - Rx^ - 2R => 2^10 - R^10C_R = 180 ....... (1) And (10 - R)r - 2R = 0 r = 2R / 10 - R r = 2(R - 10) / 10 - R + 20 / 10 - R => r = - 2 + 20 / 10 - R ....... (2) R = 8 or 5 reject equation (1) not satisfiedAt R = 8 => 2^10 - R x ^10C_R = 180 => r = 8

If the constant term in binomial expansion of 2xr 1 x2 ... | null (Mathematics)

{\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}

General term

= {}^{10}{C_R}{(2{x^2})^{10 - R}}{x^{ - 2R}}

\Rightarrow {2^{10 - R}}{}^{10}{C_R} = 180 \quad ....... (1)

And (10

-

R)r

-

2R = 0

r = {{2R} \over {10 - R}}

r = {{2(R - 10)} \over {10 - R}} + {{20} \over {10 - R}}

\Rightarrow r = - 2 + {{20} \over {10 - R}} \quad ....... (2)

R = 8 or 5 reject equation (1) not satisfied
At R = 8

\Rightarrow {2^{10 - R}}\times{}^{10}{C_R} = 180 \Rightarrow r = 8

Explanation

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