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ISI2014-DCG-35

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Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ elements, is

  1. $\begin{pmatrix} m \\ q \end{pmatrix}$
  2. $\begin{pmatrix} n \\ q \end{pmatrix}$
  3. $\begin{pmatrix} m \\ q \end{pmatrix} \times \begin{pmatrix} n \\ p-q \end{pmatrix}$
  4. $\begin{pmatrix} m \\ p-q \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
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The question actually asks how many ways are there to find a collection of $p$ elements of which it contains exactly $q$ elements from $A$ and the rest from $B$. Obviously $q\le m$ and $p\le (m+n)$.

Here, $|A|=m, ~|B|=n$

So, there are $\begin{pmatrix} m\\ q \end{pmatrix}$ ways to find $q$ elements from the set $A$

and the rest $(p-q)$ elements can be chosen from the $B$ which has $\begin{pmatrix} n\\ p-q \end{pmatrix}$ ways.

$\therefore~$The required number of ways $=\begin{pmatrix} m\\ q \end{pmatrix}\times\begin{pmatrix} n\\ p-q \end{pmatrix}$.

 

So the correct answer is C.

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