Assertion :The number of non negative integral solutions of $X_{1} + X_{2} + X_{3} + . . . . . . . . . . . . X_{n} = r i s^{r + n - 1} C_{r - 1}$ Reason: The number of ways in which $n$ identical things can be distributed among $r$ students is $^{r + n - 1} C_{r - 1}$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$X_{1} + X_{2} + . . . . . . . X_{n} = r$
To solve this means same to allow $r$ identical thing to $n$ different students $=^{n + r - 1} C r - 1$
Hence, the answer is both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

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Assertion :The number of non negative integral solutions of X1+X2+X3+............Xn=risr+n−1Cr−1 Reason: The number of ways in which n identical things can be distributed among r students is r+n−1Cr−1

Assertion :The number of non negative integral solutions of $X_{1} + X_{2} + X_{3} + . . . . . . . . . . . . X_{n} = r i s^{r + n - 1} C_{r - 1}$ Reason: The number of ways in which $n$ identical things can be distributed among $r$ students is $^{r + n - 1} C_{r - 1}$