Statement - 1: _r=0^n (r+1)^n C_r = (n+2)2^n-1 .Statement - 2: _r=0^n (r+1)^n C_r x^r = (1+x)^n + nx(1+x)^n-1 . - Mathematics MCQ AIEEE 2026 | ExamTest.live

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Statement - 1: \sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1} .Statement - 2: \sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1} .

A $Statement - 1 is false, Statement - 2 is true$
B $Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1$
C $Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1$
D $Statement - 1 is true, Statement - 2 is false$

Solution & Step-by-step Explanation

Consider Statement 2: S = \sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = \sum_{r = 0}^{n} r^{n} C_{r} x^{r} + \sum_{r = 0}^{n}^{n} C_{r} x^{r} S = x \frac{d}{d x} \sum_{r = 0}^{n}^{n} C_{r} x^{r} + (1 + x)^{n} S = x \frac{d}{d x} (1 + x)^{n} + (1 + x)^{n} = x n (1 + x)^{n - 1} + (1 + x)^{n} So, Statement 2 is true.Substituting x = 1 in Statement 2: \sum_{r = 0}^{n} (r + 1)^{n} C_{r} = n (1) (2)^{n - 1} + 2^{n} = n 2^{n - 1} + 2 \cdot 2^{n - 1} = (n + 2) 2^{n - 1} .So, Statement 1 is true and Statement 2 is the correct explanation.

Try it yourself before checking the explanation above.

Statement - 1: \sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1} .Statement - 2: \sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1} .

Statement - 1 is false, Statement - 2 is true

Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1

Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1

Statement - 1 is true, Statement - 2 is false

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