Questions: For a segment of a radio show, a disc jockey can play 5 records. If there are 7 records to select from, in how many ways can the program for this segment be arranged?

For a segment of a radio show, a disc jockey can play 5 records. If there are 7 records to select from, in how many ways can the program for this segment be arranged?

Transcript text: For a segment of a radio show, a disc jockey can play 5 records. If there are 7 records to select from, in how many ways can the program for this segment be arranged?

Solution

Solution Steps

Step 1: Understand the Problem

We are given a problem where a disc jockey can play \(n\) records from a selection of \(m\) records. We need to find out in how many ways the program for this segment can be arranged, assuming each record is distinct and order matters.

Step 2: Apply the Permutation Formula

The formula to calculate the number of ways to arrange \(n\) records out of \(m\) available is given by: \[ P(m, n) = \frac{m!}{(m-n)!} \]

Step 3: Calculate Factorials

For \(m = 7\), \(m!\) is calculated as 5040. For \(m-n = 2\), \((m-n)!\) is calculated as 2.

Step 4: Calculate Permutations

Substituting the values into the formula, we get \[ P(7, 5) = \frac{5040}{2} = 2520 \]