What if there are many groups, and not just two? For example, a test consists of six multiple-choice questions. Each question has four possible answers. How many possible answer combinations are there?
According to the product rule, if each question can be answered in four different ways, then two questions can be answered in 4 * 4 = 16 ways. Extending the rule, three questions can be answered in 4 * 4 * 4 = 64 ways. And six questions can be answered in 4 * 4 * 4 * 4 * 4 * 4 = 46 ways.
In other words, when m different events can occur in n independent ways, there are mn possible combinations!
Let us now try to solve problems that use both the sum and product rules. Val wants to go to Cleveland. She can choose from two bus services or one taxi service to head from home to downtown Dayton. From there, she can choose from two train services to head to Cleveland. How many ways are there for Val to get to Cleveland?
Val has 2 + 1 = 3 ways to reach downtown Dayton (sum rule). From there she has 2 ways to reach Cleveland, so she has 3 * 2 = 6 ways to get to Cleveland (product rule).
Val has a choice of six possible ways (combination of bus/taxi/train) to reach Cleveland.
A website is offering a date night special: pick flowers or chocolates, one movie from eight choices, and one restaurant from five choices, all for $50. How many possible date night options are there?
Use the sum rule for the or within each group: 2 options for flowers/chocolate, 8 options for movies, 5 options for restaurants.
Use the product rule for the and across groups: flowers/chocolate and movie and restaurant.
Answer: 2 * 8 * 5 = 80