The term independent of x in (1+x)m(1+(1/x))n is

Question

The term independent of x in (1+x)m(1+(1/x))n is (A)  m+n Cm (B)  m+n Cn (C)  m+n Cm-n (D) none of these

Tardigrade Admin · Accepted Answer

We have, (1+x)m(1+(1/x))n=(1+x)m((x+1/x))n =((1+x)m+n/xn)=x-n(1+x)m+n ∴ Required term independent of x= coefficient of x0 in x-n(1+x)m+n= text coefficient of xn text in (1+x)m+n = m+n Cn

Q. The term independent of $x$ in $(1 + x)^{m} (1 + \frac{1}{x})^{n}$ is

$^{m + n} C_{m}$

$^{m + n} C_{n}$

$^{m + n} C_{m - n}$

none of these

We have,
$(1 + x)^{m} (1 + \frac{1}{x})^{n} = (1 + x)^{m} (\frac{x + 1}{x})^{n}$
$= \frac{( 1 + x ) ^{m + n}}{x ^{n}} = x^{- n} (1 + x)^{m + n}$
$∴$ Required term independent of $x =$ coefficient of $x^{0}$ in
$x^{- n} (1 + x)^{m + n} = coefficient of x^{n} in (1 + x)^{m + n}$
$=^{m + n} C_{n}$

Q. The term independent of x in (1+x)m(1+x1​)n is

m+nCm​

m+nCn​

m+nCm−n​