Assertion The term independent of x in the expansion of $[x + \frac{1}{x} + 2]^{21} is^{42} C_{21}$
Reason In a binomial expansion, the middle term is independent of 𝑥 .

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Assertion is True, Reason is True but Reason is not correct explanation of Assertion

Assertion is True, Reason is False

Assertion is True, Reason is True and Reason is correct explanation of Assertion

Assertion is False, Reason is True

answer is C.

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Detailed Solution

Assertion : $= [{(\sqrt{x})}^{2} + {(\frac{1}{\sqrt{x}})}^{2} + 2]$

$= {(\sqrt{x} + \frac{1}{\sqrt{x}})}^{42}$

$n = 42$

$T_{r + 1} = {}^{n}C_{r .} x^{n - r} . a^{r}$

$=^{42} C_{r} \cdot {(\sqrt{x})}^{42 - r} \cdot {(\frac{1}{\sqrt{x}})}^{r}$

$=^{42} C_{r} \cdot {(\sqrt{x})}^{42 - r}$

By independent

$\Rightarrow 42 - 2 r = 0$

$r = 21$

∴ The independent term is $^{42} C_{21}$

Reason : The given condition not possible in all cases.

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