Addition Principle

This sub-section states the Addition Principle formally: suppose one operation can be done in $m$ ways and another operation can be done in $n$ ways, with no way common to the two operations. If ONLY ONE of these operations is to be performed, then there are $m+n$ ways to do it. The reasoning, illustrated in the parent section via the 'what to wear' tree diagram, is that when a choice is made from one exclusive alternative or the other (never both at once), the total number of possible outcomes is simply the sum of the outcomes available under each alternative — since no outcome could ever be double-counted across the two exclusive groups.

Two worked examples anchor the principle. Example 1: a restaurant offers five types of fruit juice and three types of milk shake; a customer wanting to order a single drink has 5 ways to pick a fruit juice or 3 ways to pick a milk shake, and since only one drink is chosen overall, the total is $5+3=8$ choices. Example 2: drawing one card from a standard 52-card pack, asking for the number of ways the card drawn is a spade OR a club — since the pack has 13 spade cards and 13 club cards, and no card is both a spade and a club, the count is $13+13=26$ ways. …