As a television executive, you have been given 9 shows to choose from to run during your prime time slots each week. If you have 7 time slots, how many ways can you create the schedule for the week?
Question
As a television executive, you have been given 9 shows to choose from to run during your prime time slots each week. If you have 7 time slots, how many ways can you create the schedule for the week?
Solution
This is a problem of permutations without repetition.
Step 1: Identify the number of items to choose from and the number of positions to fill. In this case, you have 9 shows to choose from and 7 time slots to fill.
Step 2: Use the formula for permutations without repetition, which is nPr = n! / (n - r)!. Here, n is the number of items to choose from (9 shows) and r is the number of positions to fill (7 time slots).
Step 3: Calculate the factorial of n, which is 9!. This means you multiply all positive integers from 1 to 9. So, 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880.
Step 4: Calculate the factorial of (n - r), which is (9 - 7)!. This means you multiply all positive integers from 1 to 2. So, (9 - 7)! = 2! = 2 * 1 = 2.
Step 5: Divide the factorial of n by the factorial of (n - r) to get the number of permutations. So, 362,880 / 2 = 181,440.
Therefore, there are 181,440 ways to create the schedule for the week.
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