One example is Latin squares and its application in design experiment. You can search in the web for more information about it and its application in everywhere.  

Also, You can see the very interesting book with name:

" I' explosion des Mathematiques " that is in French language.

Finally, I give a sweet example that maybe your audiences love it. The number of divisors of a small number is very important. it is a pure number theoretic problem. But, suppose a company want to packing its production like as chocolate or  biscuit. They prefer to use number 4, 6 or 12. Why? for better sharing the chocolates or biscuits  in a package among some friends.