Question

for a segment of a radio show, a disc jockey can play 9 records. if there are 12 records to select from, in how many ways can the program for this segment be arranged?
ways

Explanation:

Step1: Use permutation formula

The formula for permutations is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 12\) (total number of records) and \(r=9\) (number of records to play).

Step2: Calculate factorial values

\(n!=12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\) and \((n - r)!=(12 - 9)!=3!=3\times2\times1\)

Step3: Substitute into the formula

\(P(12,9)=\frac{12!}{3!}=\frac{12\times11\times10\times9\times8\times7\times6\times5\times4\times3!}{3!}\)
Cancel out the \(3!\) terms. Then \(P(12,9)=12\times11\times10\times9\times8\times7\times6\times5\times4\)
\(12\times11 = 132\), \(132\times10=1320\), \(1320\times9 = 11880\), \(11880\times8=95040\), \(95040\times7 = 665280\), \(665280\times6=3991680\), \(3991680\times5 = 19958400\), \(19958400\times4=79833600\)

Answer:

\(79833600\)