Let $X$ and $Y$ be disjoint sets containing $p$ and $q$ elements respectively, and let the set $Z$ be defined as $Z\bigcup Y$. Find the number of subsets $S$ of $Z$ that contains $m$ elements and also has the property that $S\bigcap X$ has $n$ elements.

What I've tried is:

Total elements in $Z$ is $p+q$, so subsets containing $m$ elements is $C(p+q,m)m!$ where $C$ denotes combination.

How to include $n $ elements interesection ?


  • $\begingroup$ Hint: first choose the $n$ elements of $S$ which are in $X$, and then choose the $m-n$ elements which are in $Y$. $\endgroup$ – Mike Earnest Feb 7 at 17:33

To construct such a subset, we must take $n$ elements fom $X$ and $m-n$ elements from $Y$, so the number of such subsets is

$$\binom{|X|}{n} \cdot \binom{|Y|}{m-n}$$

  • $\begingroup$ Thanks a lot, I got it! $\endgroup$ – Henam Feb 7 at 17:44

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