Q.A fixed-price meal lets a customer choose one of 3 soups or skip the soup course entirely, then choose one of 5 main courses and one of 2 desserts. In how many ways can a customer choose a meal?

The Fundamental Principle of Counting is the basic toolkit for counting outcomes without listing them one by one, and it has two complementary halves. The Addition Principle applies when a single choice is made from among several mutually exclusive alternatives: if one operation can be carried out in $m$ ways and a second, entirely separate operation can be carried out in $n$ ways, and only ONE of the two operations is actually performed, then the total number of ways is $m+n$. The word 'or' in a problem is the usual signal for this — choosing a drink that is EITHER a fruit juice OR a milkshake. The Multiplication Principle applies when two (or more) operations are carried out one after another, in sequence, as part of the SAME overall task: if the first can be done in $m$ ways and, independently of that choice, the second can be done in $n$ ways, then together they can be done in $m\times n$ ways. The word 'and' is the usual signal — choosing a shirt AND a matching pair of trousers. Both principles extend naturally from two activities to any finite number of activities, simply by adding …