Fundamental principles of counting

This section lays the groundwork for every counting technique in the chapter: the two Fundamental Principles of Counting (Addition and Multiplication), introduced through the visual device of a tree diagram.

A tree diagram is a branching picture that shows how a set splits into disjoint (non-overlapping) subsets — an alternative to a Venn diagram, better suited to situations with several successive layers of choice. The chapter's first tree diagram (Fig. 3.1) illustrates this with 'School Games': the root splits into two branches, Indoor and Outdoor, and each of those splits further into the specific games available under it (Chess, Carrom, Table Tennis under Indoor; Cricket, Volleyball, Basketball, Badminton under Outdoor). Every leaf of the tree is one distinct game, reached by exactly one path down from the root — this visual structure is what the Addition and Multiplication Principles turn into arithmetic.

The Addition Principle is introduced through a boy deciding what to wear (Fig. 3.2): he has 3 T-shirts (white, green, blue) and 4 shirts (red, green, yellow, orange), and wants to wear ONE garment. The tree diagram shows 4 branches under 'Shirt' and 3 branches under 'T-Shirt' — 7 leaves in total — and the text draws out the general rule directly from this: the boy can choose from the 4 shirts OR from the 3 T-shirts, and since only one is worn, the total number of ways is $3+4=7$, obtained by ADDING the counts of the two alternatives. This is stated formally as the Addition Principle: if one operation can be done in $m$ ways and another, entirely separate operation can be done in $n$ ways, with no way common to both, then if only ONE of these operations is to be performed, there are $m+n$ ways to do it. Two worked examples reinforce this: (i) a restaurant offering 5 types of fruit juice and 3 types of milkshake, where a customer selecting exactly one drink has $5+3=8$ choices; and (ii) drawing one card from a 52-card pack and asking for the number of ways the card is a spade OR a club — since there are 13 of each and no card is both, the count is $13+13=26$. A closing Note flags the linguistic signal for this principle: the word 'OR' in a problem statement suggests addition. …