A few simpler problems

This lecture will discuss some issues related to the introduction of the Combinatorics before high school students. Combinatorics is naturally invoked in problems when the question is how many choices there are for some event to occur, how many objects a set contains, etc. Students find these problems entertaining and the subject is fairly easy to introduce through games and competitions in class.

Problem1. There are five different teacups and three different tea saucers in the ``Tea Party'' store. How many ways are there to buy a cup and a saucer.
Solution. First, let us choose a cup. Then, to complete the set, we can choose any of three saucers. Thus we have 3 different sets containing the chosen cup. Since there are 5 cups, we have 15 different sets ( $3\cdot5=15$).

Problem2. There are also in the ``Tea party'' store four different teaspoons. How many ways are there to buy a set of a cup, a saucer and a spoon?
Solution. Let us start with any of the 15 sets from the previous problem. There are four different ways to complete it by choosing a spoon. Therefore, the number of all possible sets is $15\cdot4=60$.

Methodological Remark. The main goal the teacher should pursue during a discussion of these problems is making the students understand when we must add the numbers and when we multiply them. Of course, many problems should be presented. Some possible subjects are shopping, traffic maps, arrangement of objects, etc.