Problem 45 - Algebra and Trigonometry

Question

In a race in which six automobiles are entered and there are no ties, in how many ways can the first three finishers come in?

Step-by-Step Solution

Answer

The first three finishers can come in a total of 120 different ways.

Step 1: Understanding the problem

In this problem, we are asked to find the number of ways the first three finishers can occur from a total of six automobiles. The order of finishing matters in this problem, so it's a permutation problem.

Step 2: Using the fundamental principle of counting

Using the rule of multiplication referred to as the fundamental principle of counting, if an event can occur in n ways and a separate event can occur in m ways, then the number of ways the two events can occur is n multiplied by m. In this problem, the first event is the first car crossing the finish line, which can happen in 6 ways, as there are 6 cars. The second event is the second car crossing, which can happen in 5 ways, as one car has already finished. The third event is the third car crossing can happen in 4 ways as two cars have already finished.

Step 3: Calculation

The total number of ways the first three finishers can occur is the product of the number of ways each event can occur. Therefore, we calculate the total number of ways as \(6 * 5 * 4 = 120\) ways.