Summary
Highlights
The fundamental principle of counting states that if one event can occur in N1 ways, a second event in N2 ways, and a third event in N3 ways, and so on, then these events can occur in N1 * N2 * N3 * ... ways. This principle is similar to using tree diagrams to find the number of possible outcomes.
Sarah Geronimo is planning a concert tour in three cities: Manila, Cebu, and Davao. To find the number of ways she can arrange her tour schedule, we multiply the number of choices for each stop: 3 (for the first stop) * 2 (for the second stop) * 1 (for the third stop) = 6 possible ways.
When tossing a coin thrice, each toss has two possible outcomes (head or tail). Therefore, the total number of possible outcomes is 2 * 2 * 2 = 8.
A bus has six vacant seats and two additional passengers enter. The first passenger has 6 choices, and the second passenger has 5 choices. So, there are 6 * 5 = 30 ways they can be seated.
A clown has four different colors of balloons, and two children will receive one balloon each. The first child has 4 choices, and the second child has 3 choices. Thus, there are 4 * 3 = 12 ways this can be done.
In a math contest with 15 contestants, there are first, second, and third prizes. The first prize can be awarded in 15 ways, the second in 14 ways, and the third in 13 ways. The total number of ways to award the prizes is 15 * 14 * 13 = 2730 ways.
A date night special offers one movie from four choices, one restaurant from six choices, and either flowers, chocolate, or wine (3 choices). The total number of possible date night options is 4 * 6 * 3 = 72 options.
Automobile license plates display three letters followed by two numbers. If repetition of letters is not allowed, there are 26 choices for the first letter, 25 for the second, and 24 for the third. For numbers (0-9), there are 10 choices for the first and 10 for the second. So, the total is 26 * 25 * 24 * 10 * 10 = 1,560,000 possible plates.
In a family of five children, all five children are seated in a row, with the condition that the girls must sit on the end chairs. This example is presented as an exercise with the final calculation of 2 * 3 * 2 * 1 * 1 = 12 ways.