Questions: As a television executive, you have been given 13 shows to choose from to run during your prime time slots each week. If you have 9 time slots, how many ways can you create the schedule for the week?

Solution Steps

To solve this problem, we need to determine the number of ways to choose and arrange 9 shows out of 13 available shows. This is a permutation problem because the order in which the shows are scheduled matters. We can use the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange.

The problem is a permutation problem where we need to determine the number of ways to arrange 9 shows out of 13 available shows. The order of arrangement matters, which is why we use permutations.

The formula for permutations is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] where \( n = 13 \) (total number of shows) and \( r = 9 \) (number of time slots).

Substitute the values into the permutation formula: \[ P(13, 9) = \frac{13!}{(13-9)!} = \frac{13!}{4!} \]

Calculate the factorials: \[ 13! = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Now, compute the permutation: \[ P(13, 9) = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5}{24} \]

Perform the division to find the number of ways to schedule the shows: \[ P(13, 9) = 259459200 \]

Final Answer