Question

Assertion :${\sum }_{r=0}^{n}r({}^n{C}_r{)}^2=n({}^{2n-1}{C}_{n-1})$ Reason: ${\sum }_{r=0}^n({}^n{C}_r{)}^2={}^{2n}{C}_n$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

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

${\sum }_{r=0}^{n}r{\left({}^n{C}_r\right)}^2={\left({}^n{C}_1\right)}^2+2\times {\left({}^n{C}_2\right)}^2+3{\left({}^n{C}_3\right)}^2+....+n{\left({}^n{C}_n\right)}^2$

Now, ${(1+x)}^n{=}^n{C}_0+x\left({}^n{C}_1\right)+x^2\left({}^n{C}_2\right)+x^3\left({}^n{C}_3\right)+...+x^n\left({}^n{C}_n\right)$

Taking derivative on both sides,

$n{(1+x)}^{n-1}{=}^n{C}_1+2x\left({}^n{C}_2\right)+3x^2\left({}^n{C}_3\right)+...+{nx}^{n-1}\left({}^n{C}_n\right)$

Also, $(x+1{)}^n{=}^n{C}_0{x}^n{+}^n{C}_1{x}^{n-1}+....{+}^n{C}_n{x}^0$

Now, ${\sum }_{r=0}^n{\left({}^n{C}_r\right)}^2=$ coefficient of $x^n$ in $(1+x{)}^{2n}=$ ${}^{2n}{C}_n$

Similarly, ${\sum }_{r=0}^{n}r{\left({}^n{C}_r\right)}^2=$ coefficient of $x^{n-1}$ in $n(1+x{)}^{n-1}\times (1+x{)}^n=n\times (1+x{)}^{2n-1}=$ $n{\times }^{2n-1}{C}_{n-1}$

hence, assertion and reason are both correct but reason is not the correct explanation for assertion.