Problem 43 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) For a segment of a radio show, a disc jockey can play 7 songs. If there are 13 songs to select from, in how many ways can the program for this segment be arranged?

Step-by-Step Solution

Answer

The disc jockey can arrange the program in 3,315,312 different ways.

Step 1: Understand the Problem

The disc jockey needs to choose 7 songs out of 13 available and the order in which these songs are played matters. So, we have a permutation problem here. n represents number of total items available which is 13 in this case and r represents number of items we want to choose which is 7 here.

Step 2: Apply Permutations Formula

To solve the problem, we use the Permutations formula which is \(_{n} P_{r} = \frac{n!}{(n-r)!}\). Here n is 13 and r is 7. So, we substitute these values into the formula. We get \(_{13} P_{7} = \frac{13!}{(13-7)!}\)

Step 3: Calculate Factorials

Next, we calculate the factorials. Factorial of a number n (denoted by n!) is the product of all positive integers less than or equal to n. So, we calculate 13! and 6!.

Step 4: Calculate Final Result

Finally, we divide 13! by 6! to find the number of ways the disc jockey can arrange the songs. That will give us the final answer.