Assume that the producer has 8 skits and 5 commercials available.



In how many ways can the producer create a television show?

How many skits and commercials does the show have to have? Can the commercials be consecutive?

Are you only interested in the number of orders in which they all can be arranged?

To determine the number of ways the producer can create a television show, we need to consider the number of skits and commercials.

The producer has 8 skits and 5 commercials available.

The skits and commercials can be arranged in any order, so we need to find the number of permutations.

To calculate the number of permutations, we can use the formula for permutations:

P(n, r) = n! / (n - r)!

Where n is the total number of items and r is the number of items chosen.

In this case, we want to find the number of permutations of 8 skits and 5 commercials, so n = 13 (8 + 5) and r = 13.

P(13, 13) = 13! / (13 - 13)!
= 13! / 0!
= 13! / 1
= 13!

Therefore, the producer can create a television show in 13! (factorial) ways.

To figure out the number of ways the producer can create a television show, we need to use the concept of combinations.

Combinations are used when selecting items from a larger set without considering the order in which they are chosen. In this case, the skits and commercials can be combined in any order to create a television show.

The formula for calculating combinations is:
nCr = n! / (r!(n-r)!)

Where "n" is the total number of items and "r" is the number of items selected.

In this case, we have 8 skits and 5 commercials, so the total number of items is 8 + 5 = 13.

To find the number of ways the producer can create a television show, we need to calculate 13C8 (the number of ways to choose 8 items from a set of 13).

Plugging in the values into the combination formula, we get:
13C8 = 13! / (8!(13-8)!)
= 13! / (8!5!)

Now, let's calculate the factorial values:
13! = 13 * 12 * 11 * 10 * 9 * 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1

By canceling out some terms, we can simplify the equation:
13! / (8!5!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1)

Evaluating this expression, we get:
(13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1,287

Therefore, the producer can create a television show in 1,287 different ways using the given skits and commercials.