Assertion :The number of terms in the expansion ${(x + (1 / x) + 1)}^{n}$ are $2 n + 1$ . Reason: The number of terms in the expansion ${(n \sum r = 1 a_{r})}_{r}^{n}$ is $^{n + r - 1} C_{r - 1}$ .

Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,

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Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,

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Assertion is true but Reason is false,

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Assertion is false but Reason is true.

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Solution

The correct option is B Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
${(x + (\frac{1}{x}) + 1)}^{n} = \frac{{(x^{2} + x + 1)}^{n}}{x^{n}} = \frac{a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{2 n} x^{2 n}}{x^{n}} = \frac{a_{0}}{x^{n}} + \frac{a_{1}}{x^{n - 1}} + \frac{a_{2}}{x^{n - 2}} + . . . + \frac{a_{n - 1}}{x} + a_{n} + a_{n + 1} x + a_{n + 2} x^{2} + . . . + a_{2 n} x^{n}$

Hence, ${(x + (\frac{1}{x}) + 1)}^{n}$ has $2 n + 1$ terms.
Now, ${(\sum_{r = 1}^{n} a_{r})}^{n}$ has $^{n + r - 1} C_{r}$ terms, where $a_{0}, a_{1,} a_{2} . . . a_{r}$ are indepnendent variables.

So, the Assertion and reason are both True but the reason is true only if $a_{0}, a_{1,} a_{2} . . . a_{r}$ are independent variables but $x$ and $1 / x$ are not independent variables. They vary with each other.

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