Assertion :The coefficient of $T_{r + 1}$ in the expansion ${(1 + x)}^{n + 1} =$ sum of the coefficients of $T_{r}$ & $T_{r + 1}$ in ${(1 + x)}^{n}$ Reason: $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

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 A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
Writing the general term for $(1 + x)^{n}$
$T_{r} =^{n} C_{r - 1} x^{r - 1}$
$T_{r + 1} =^{n} C_{r} x^{r}$
$T_{r} + T_{r + 1}$
$=^{n} C_{r - 1} +^{n} C_{r}$ ...(only the coefficients)
$=^{n + 1} C_{r}$ ...(i)
Writing the general term for $(1 + x)^{n + 1}$
$T_{r + 1} =^{n + 1} C_{r} x^{r}$ ...(ii)
Hence coefficients of ii and i are same.
Hence assertion is true and the reason is the correct explanation of the assertion.

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