Prove that:

$\left(\begin{array}{c}n+m\\ r\end{array}\right)=\left(\begin{array}{c}n\\ 0\end{array}\right)\left(\begin{array}{c}m\\ r\end{array}\right)+\left(\begin{array}{c}n\\ 1\end{array}\right)\left(\begin{array}{c}m\\ r-1\end{array}\right)+..........+\left(\begin{array}{c}n\\ r\end{array}\right)\left(\begin{array}{c}m\\ 0\end{array}\right)$

Hint: Consider a group of $n$men and $m$women. How many groups of size $r$are possible?

Here, it is given that

No. of men $=n$

No. of women $=m$

Total no. of members in group $=n+m$

A sub group of $r$members is to be formed.

The possible combinations of a group of $r$members are -

Case 1: Selecting role="math" localid="1647930590507" $r$women from the group of women and role="math" localid="1647930645718" $0$men from the group of men role="math" localid="1647930941948" $={}^{n}C_0\times {}^{m}C_r$

Case 1: Selecting $r-1$women from the group of women and $1$men from the group of men$={}^{n}C_1\times {}^{m}C_{r-1}$

Case 1: Selecting $r-2$women from the group of women and $2$men from the group of men$={}^{n}C_2\times {}^{m}C_{r-2}$

.....

.....

Case 1: Selecting $1$women from the group of women and $r-1$men from the group of men$={}^{n}C_{r-1}\times {}^{m}C_1$

Case 1: Selecting $0$women from the group of women and$r$ men from the group of men$={}^{n}C_r\times {}^{m}C_0$

The possible number of ways of selecting $r$members from the group of $n+m$members will be

localid="1647931140939" $\left(\begin{array}{c}n+m\\ r\end{array}\right)={}^{n}C_0\times {}^{m}C_r+{}^{n}C_1\times {}^{m}C_{r-1}+{}^{n}C_2\times {}^{m}C_{r-1}+..........+{}^{n}C_{r-1}\times {}^{m}C_1+{}^{n}C_r\times {}^{m}C_0$

Hence it is proved that

$\left(\begin{array}{c}n+m\\ r\end{array}\right)=\left(\begin{array}{c}n\\ 0\end{array}\right)\left(\begin{array}{c}m\\ r\end{array}\right)+\left(\begin{array}{c}n\\ 1\end{array}\right)\left(\begin{array}{c}m\\ r-1\end{array}\right)+..........+\left(\begin{array}{c}n\\ r\end{array}\right)\left(\begin{array}{c}m\\ 0\end{array}\right)$