How many ways can twelve 30-second commercials be arranged to be shown in a one-hour television

How many ways can twelve 30-second commercials be arranged...

Key Concepts

Permutations

Permutations refer to the arrangement of items in a specific order. When the order matters, as in arranging a sequence of commercials or objects, the number of possible arrangements is calculated using permutation methods. This concept is foundational in combinatorics, where counting the different sequences is essential for solving arrangement problems.

Factorial

The factorial operation is used to compute the product of all positive integers up to a given number, and it plays a central role in calculating permutations. Denoted as n!, the factorial of n gives the number of ways to arrange n distinct items in order, making it a crucial tool in combinatorial problems involving arrangements.

Transcript

00:02 Okay, this question has us considering commercial arrangements for a set of six restaurants and five stores.

00:10 So the first scenario they give us in part a is if we don't have to do any restrictions on the order.

00:18 And to do that, well, it's just arranging six plus five, which is 11 commercials.

00:24 And how many ways can we order 11 things? well, for our first commercial, we have 11 choices.

00:34 For our second commercial, we have 10 choices.

00:37 Then we have 9, 8, 7, all the way down.

00:43 So this is equal to 11 factorial, which is 399168 ,000.

00:58 Different orders to put our commercials in.

01:01 Then for part b, it says we need to group the stores together and the restaurants.

01:10 Together.

01:14 So the way i think about this one is we have our first choice, which is what i'll call the block order, and that's just which one we choose first.

01:35 Do we want the store block first, or do we want the restaurant block first? then we have to multiply by the store number and the restaurant number of arrangements.

02:05 Because if we group them together with the stores, there's five.

02:09 So we have five factorial ways to order the store commercials, six factorial ways to order the restaurant commercials.

02:21 So multiplying these two together would be how many different orders we can have with one block configuration, but we could do the stores first or the store's second.

02:31 So we have two factorial ways to arrange the block of what order the commercials show up in because we have one order where the restaurants are first and one order where the stores are first so we have to multiply the total by two factorial and if we do this this works out to be one seven to eight zero zero then for part c it says the the restaurants and stores most alternate...

Please give Ace some feedback