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Chapter 11: Problem 43

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 number of ways the program for this segment can be arranged is the result obtained after dividing the value of \(13!\) by the value of \(6!\).

Step by step solution

01

Understanding Permutations

A permutation is an arrangement of objects in a specific order. In this case, we are arranging songs on a radio program. The number of permutations of n distinct objects taken r at a time is given by the formula \(nPr = n! / (n - r)!\), where 'n' is the total number of options, 'r' is the number of options chosen, and '!' denotes a factorial, meaning the product of all positive integers up to that number.
02

Applying the Permutations Formula

Applying this formula to the disc jockey situation, we determine 'n' to be the total number of songs (13) and 'r' to be the number of songs chosen for the segment (7). Substituting these values, we have \(13P7 = 13! / (13 - 7)!\).
03

Calculating Factorials and Solving the Equation

Now it's time to compute the factorials. Factorial of 13, denoted as \(13!\), is the product of all positive integers up to 13. Similarly, factorial of 6 i.e., \((13 - 7)!\), is the product of all positive integers up to 6. You can calculate these using a calculator. Once we have the values, we can divide the value of \(13!\) by the value of \(6!\) to get the answer.

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