Question

If r > p > q, the number of different selections of p + q things taking r at time, where p things are identical and q other things are identical, is

• A. $p+q-r$
• B. $p+q-r+1$
• C. $r-p-q+1$
• D. none of these

Answer 1

The correct option is B $p+q-r+1$

Let $A$ be the set containing $p$ identical things and $B$ be the set containing $q$ identical things. The number of selections of $r$ things from set $A$ and set $B$ is similar to finding the number of solutions of
$x_1+x_2=r$ , where $0\le x_1\le p$ and $0\le x_2\le q$.
Since, $r>p>q$, the greatest value of $x_1$ will be $p$ and the greatest value of $x_2$ will be $q$.
Hence, $x_1$ will take values from $(r-q)$ to $p$. Hence, the number of values $x_1$ can take will be $p-(r-q)+1=p+q-r+1$
This is the same as the number of selections of $r$ things from $p+q$ things , where $p$ things are identical and $q$ other things are identical.