If xm occurs in the expansion of (x + 1/x2)2n, then the coefficient of xm is

Question

If xm occurs in the expansion of (x + 1/x2)2n, then the coefficient of xm is (A) ((2n)!/(m)!(2n-m)!) (B) ((2n)!3!3!/(2n-m)!) (C) ((2n)!/((2n-m)3)!((4n+m)3)! (D) none of these

Tardigrade Admin · Accepted Answer

Tr+1 = 2nCrx2n-r((1/x2))r= 2nCrx2n-3r This contains xm. If 2n -3r =m, then r=(2n-m/3) ⇒ Coefficient of xm = 2nCr r=(2n-m/3) =(2n!/(2n-r)!r!)=(2n!/(2n-(2n-m)3)!((2n-m)3)! =(2n!/((4n+m)3)!((2n-m)3)!

Q. If $x^{m}$ occurs in the expansion of $(x + 1/ x^{2})^{2 n}$ , then the coefficient of $x^{m}$ is

$\frac{( 2 n ) !}{( m ) ! ( 2 n - m ) !}$

$\frac{( 2 n ) ! 3 ! 3 !}{( 2 n - m ) !}$

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

none of these

Q. If xm occurs in the expansion of (x+1/x2)2n, then the coefficient of xm is

(m)!(2n−m)!(2n)!​

(2n−m)!(2n)!3!3!​

(32n−m​)!(34n+m​)!(2n)!​

none of these

Tr+1​=2nCr​x2n−r(x21​)r=2nCr​x2n−3r This contains xm. If 2n−3r=m, then r=32n−m​ ⇒ Coefficient of xm=2nCr​ r=32n−m​ =(2n−r)!r!2n!​=(2n−32n−m​)!(32n−m​)!2n!​ =(34n+m​)!(32n−m​)!2n!​

Q. If $x^{m}$ occurs in the expansion of $(x + 1/ x^{2})^{2 n}$ , then the coefficient of $x^{m}$ is

$\frac{( 2 n ) !}{( m ) ! ( 2 n - m ) !}$

$\frac{( 2 n ) ! 3 ! 3 !}{( 2 n - m ) !}$

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