To determine the number of ways the program for the segment can be arranged, we must calculate the number of permutations of 9 records from a set of 14.
A permutation is an arrangement of items where order matters. When selecting r items from a set of n without replacement, the number of permutations is given by the formula:
\[ P(n, r) = n! / (n - r)! \]
where \( n! \) (n factorial) represents the product of all positive integers from 1 to n, and \( (n - r)! \) is the factorial of the difference between n and r.
Now let's solve the problem step by step.
Step 1: Identify n and r.
- n (total number of different records to select from) = 14
- r (number of records to arrange) = 9
Step 2: Apply the permutation formula.
\[ P(14, 9) = \frac{14!}{(14 - 9)!} = \frac{14!}{5!} \]
Step 3: Calculate factorials and simplify.
\( 14! \) (14 factorial) means the product of all positive integers from 1 to 14:
\[ 14! = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
\( 5! \) (5 factorial) is the product of all positive integers from 1 to 5:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]
Step 4: Divide 14! by 5! to find the number of permutations.
Since \( 5! \) is a factor in both the numerator and the denominator, we can simplify the calculation by canceling out the common factors.
\[ P(14, 9) = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \]
The \( 5 \times 4 \times 3 \times 2 \times 1 \) in the denominator will cancel out the \( 5 \times 4 \times 3 \times 2 \times 1 \) in the numerator, so we're left with:
\[ P(14, 9) = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \]
Step 5: Calculate the product of the remaining factors to get the final answer.
\[ P(14, 9) = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 = 1,037,836,800 \]
So there are 1,037,836,800 different ways the disc jockey can arrange the 9 records for the segment of the radio show.