If {x^m} occurs in the expansion of {left( {x | vedclass

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(C) The general term in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}}$ is given by: ${T_{r + 1}} = {}^{2n}{C_r}{x^{2n - r}}{\left( {

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$\frac{{(2n)!}}{{\left( {\frac{{2n - m}}{3}} \right)!\,\left( {\frac{{4n + m}}{3}} \right)!}}$

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(C) The general term in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}}$ is given by: ${T_{r + 1}} = {}^{2n}{C_r}{x^{2n - r}}{\left( {{x^{ - 2}}} \right)^r} = {}^{2n}{C_r}{x^{2n - 3r}}$ For the term containing ${x^m}$,we set the exponent equal to $m$: $2n - 3r = m \implies 3r = 2n - m \implies r = \frac{{2n - m}}{3}$ The coefficient of ${x^m}$ is ${}^{2n}{C_r} = \frac{{(2n)!}}{{r!(2n - r)!}}$ Substituting $r = \frac{{2n - m}}{3}$: $2n - r = 2n - \frac{{2n - m}}{3} = \frac{{6n - 2n + m}}{3} = \frac{{4n + m}}{3}$ Thus,the coefficient is $\frac{{(2n)!}}{{\left( {\frac{{2n - m}}{3}} \right)!\,\left( {\frac{{4n + m}}{3}} \right)!}}$.

If ${x^m}$ occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is

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