Question

Assertion :$\frac{1}{m!}{C}_0+\frac{n}{(m+1)!}{C}_1+\frac{n(n-1)}{(m+2)!}{C}_2+.....+\frac{n(n-1)...2.1}{(m+n)!}{C}_n=\frac{(m+n+1)(m+n+2)...(m+2n)}{(m+n)!}$ Reason: for $r\ge 0$
$\left(\begin{array}{c}m\\ r\end{array}\right)C_0+\left(\begin{array}{c}m\\ r-1\end{array}\right)C_1+......\left(\begin{array}{c}m\\ 0\end{array}\right)C_r=\left(\begin{array}{c}m+n\\ r\end{array}\right)$

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

$\left(\begin{array}{c}m\\ r\end{array}\right)C_0+\left(\begin{array}{c}m\\ r-1\end{array}\right)C_1+......\left(\begin{array}{c}m\\ 0\end{array}\right)C_r$
$=$ number of ways of slecting $r$ persons out of $m$ men and $n$ women
$=\left(\begin{array}{c}m+n\\ r\end{array}\right)$
We can write $L.H.S$ of $\left(1\right)$ as
$\frac{n!}{(m+n)!}[\frac{(m+n)!}{m!n!}{C}_0+\frac{(m+n)!}{(m+1)!(n-1)!}{C}_1+.....\frac{(m+n)!}{(m+n)!}{C}_n]$
$=\frac{n!}{(m+n)!}\left(\begin{array}{c}m+n+n\\ n\end{array}\right)$
$=\frac{n!}{(m+n)!}\frac{(m+2n)!}{n!(m+n)!}=R.H.S$ of $\left(1\right)$