Question

for a segment of a radio show, a disc jockey can play 6 records. if there are 8 records to select from, in how many ways can the program for this segment be arranged? ways

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 6 records out of 8. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 8 \) (total records) and \( r = 6 \) (records to arrange).

Step2: Substitute values into the formula

Substitute \( n = 8 \) and \( r = 6 \) into \( P(n, r)=\frac{n!}{(n - r)!} \). First, calculate \( n-r=8 - 6=2 \). So we have \( P(8, 6)=\frac{8!}{2!} \).

Step3: Expand the factorials

We know that \( n!=n\times(n - 1)\times\cdots\times1 \). So \( 8! = 8\times7\times6\times5\times4\times3\times2! \) and \( 2! = 2\times1 \). Then \( \frac{8!}{2!}=\frac{8\times7\times6\times5\times4\times3\times2!}{2!} \). The \( 2! \) terms cancel out.

Step4: Calculate the product

Now calculate \( 8\times7\times6\times5\times4\times3 \). \( 8\times7 = 56 \), \( 56\times6 = 336 \), \( 336\times5 = 1680 \), \( 1680\times4 = 6720 \), \( 6720\times3 = 20160 \).

Answer:

20160