Statement - 1: Σ r = 0 ⁿ (r + 1) ⁿ C r = (n + 2)2ⁿ⁻¹. Statement - 2: Σ r = 0 ⁿ (r + 1) ⁿ C r x^(r) = (1 + x)ⁿ + nx (1 + x)ⁿ⁻¹.

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

(A) Statement - 1 is false, Statement - 2 is true

(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

Binomial Theorem and Its Simple Applications: JEE Main 2008 Mathematics question

JEE Main 2008 · Annual, Paper 1 · Q2970

JEE Main 2008Annual, Paper 1MathematicsMediumBinomial Theorem and Its Simple Applications131× since 2002

Statement - 1 :

\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.}

Statement - 2 :

\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( {1 + x} \right)}^{n - 1}}.}

JEE Main 2008 · Annual, Paper 1 · Q2970

JEE Main 2008Annual, Paper 1MathematicsMediumBinomial Theorem and Its Simple Applications131× since 2002

Statement - 1 :

\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.}

Statement - 2 :

\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( {1 + x} \right)}^{n - 1}}.}