Question

Assertion :For $n\in N$, let $C_r=\left(\begin{array}{c}n\\ r\end{array}\right)$, $0\le r\le n$

${\sum }_{r=1}^n\left(r\right)C_r^2=n\left(\begin{array}{c}2n-1\\ n-1\end{array}\right)$

Reason: ${\sum }_{r=1}^n{C}_r^2=\left(\begin{array}{c}2n\\ n\end{array}\right)$

• 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. Assertion is incorrect but Reason is correct

Answer 1

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

For truth of reason

We have $C_0^2+C_1^2+C_2^2+...+C_n^2=\sum _{r=0}^n\left({}^n{C}_r\right)\left({}^n{C}_r\right)$

$=\sum _{r=0}^n\left({}^n{C}_r\right)\left({}^n{C}_{n-r}\right)[\because {}^n{C}_r={}^n{C}_{n-r}]$

$=$ number of ways of choosing $n$ persons out of $n$ men and $n$ women

$=$ number of ways of choosing $n$ persons out of $2n$ persons

$={}^{2n}{C}_n$

For $r\ge 1,r{C}_r=r\frac{n!}{r!(n-r)!}=n\left(\begin{array}{c}n-1\\ r-1\end{array}\right)$

$\therefore \sum _{r=1}^{n}r{C}_r=n\sum _{r=1}^n\left(\begin{array}{c}n-1\\ r-1\end{array}\right)=n\left(\begin{array}{c}n+n-1\\ n-1\end{array}\right)=n\left(\begin{array}{c}2n-1\\ n-1\end{array}\right)$

Hence, both reason and assertion are correct.