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Chapter 8: Problem 43

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?

Short Answer

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

Step by step solution

01

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.
02

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 \(_{13} P_{7} = \frac{13!}{(13-7)!}\)
03

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!.
04

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.

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