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 to choose 6 shows to run on your network, how many ways can you choose which shows to pick up?
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 to choose 6 shows to run on your network, how many ways can you choose which shows to pick up?
Solution
This is a combination problem. You are choosing 6 shows out of 9, and the order in which you choose them does not matter.
The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of options, k is the number of options chosen, and "!" denotes a factorial, which means multiplying all positive integers up to that number.
Here, n = 9 (the total number of shows) and k = 6 (the number of shows you are choosing).
So, the calculation would be:
C(9, 6) = 9! / [6!(9-6)!] = 9! / [6!3!] = (987654321) / [(654321)(321)] = (987) / (321) = 84
So, there are 84 different ways you can choose 6 shows from 9 to run on your network.
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