A market research group conducted a survey of $1000$consumers and reported that 720 consumers liked product A and $450$ consumers liked product B. What is the least number that must have liked both products ?

Let U be the set of consumers questioned, X the set of consumers who liked the product A and Y, the set of consumers who liked the product B. Then $\left(\mathrm{U}\right)=1000$

$\begin{array}{l}\mathrm{n}\left(\mathrm{X}\right)=720,\mathrm{n}\left(\mathrm{Y}\right)=450\\ \mathrm{n}(\mathrm{X}\cup \mathrm{Y})=\mathrm{n}\left(\mathrm{X}\right)+\mathrm{n}\left(\mathrm{Y}\right)-\mathrm{n}(\mathrm{X}\cap \mathrm{Y})=1170-\mathrm{n}(\mathrm{X}\cap \mathrm{Y})\\ \therefore \mathrm{n}(\mathrm{X}\cap \mathrm{Y})=1170-\mathrm{n}(\mathrm{X}\cup \mathrm{Y})\end{array}$

Clearly, $\mathrm{n}(\mathrm{X}\cap \mathrm{Y})\text{ is least when }\mathrm{n}(\mathrm{X}\cup \mathrm{Y})$ is maximum.

Now, $\mathrm{X}\cup \mathrm{Y}\subset \mathrm{U}$

$\therefore \mathrm{n}(\mathrm{X}\cup \mathrm{Y})\le \mathrm{n}\left(\mathrm{U}\right)=1000$

$\therefore$the maximum value of  $n(\mathrm{X}\cup \mathrm{Y})\text{ is }1000$