Problem 44 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

Step-by-Step Solution

Answer

There are 6840 ways the three best movies can be chosen and ranked.

Step 1: Understand the notation

\(_{n}P_{r}\) is the notation for permutations of \(n\) things taken \(r\) at a time. Here, \(n = 20\) (the total number of movies seen) and \(r = 3\) (the number of movies to be ranked).

Step 2: Apply the permutation formula

The formula for permutations is \(_{n} P_{r} = \frac{n!}{(n-r)!}\). Apply this formula with the given values of \(n\) and \(r\). So, \(_{20} P_{3} = \frac{20!}{(20-3)!}\).

Step 3: Solve the factorial

Factorial means multiplying all positive integers up to that number. To tackle the factorials, realize that \(20! = 20 * 19 * 18 * 17!\) and in the denominator we have \(17!\) these cancel out leaving \(20 * 19 * 18\) in the numerator.

Step 4: Final calculation

Perform the multiplication: \(20 * 19 * 18 = 6840\).