First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

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

$m + {nC}_{n + 1}$
C

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

None of these

Solution

$Given, (1 + x)^{m} \cdot {(1 + \frac{1}{x})}^{n}$
To find Term independent of x
coefficient of x⁰ in $\{(1 + x)^{m} \cdot {(\frac{x + 1}{x})}^{n}\}$
$\Rightarrow$ coefficient of x⁰ in $\{\frac{(1 + x)^{m + n}}{x^{n}}\}$
$\Rightarrow$ Coefficient of xⁿ in $(1 + x)^{m + n}$
$\Rightarrow$ Coefficient of xⁿ in $\{^{m + n} C_{r} x^{r}\}$
$\Rightarrow^{m + n} C_{n}, put r = n$
.'. Term independent of. x $=^{m + n} C_{n} {or}^{m + n} C_{m}$

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