Combinatorics Examples in Real-Life

For the following set of problems determine what part of Combinatorics we need to use and apply the appropriate formula. Have in mind that there might be more than one correct approach to some (or all) of these questions.

PROBLEM 1

Imagine you are working at an office and have 5 tasks labelled as “Critical” in Jira to complete by the end of the day. In how many ways can you complete the said tasks before the day ends?

** “Jira” is a Project Management software which allows you to create tasks and label them depending on their importance. “Critical” is the highest level of importance and no task with lower-level can be started once such a task is initiated.

Solution: —

We need to arrange all 5 tasks; hence, we are looking for the number of Permutation between 5 elements. Thus, we have 5! = 120 ways of completing our assignments.

PROBLEM 2

Imagine your company is trying to gain customers by running an online ad campaign. The idea is to focus on a certain demographic which frequently uses social media. Your campaign will run ads on Facebook, Messenger, Instagram, Twitter, and Reddit (in total = 5). Your graphical designers have created 8 different version of the banner you can use. Based on this information: —

1. Calculate how many different options you have for the entire campaign, assuming you want to use a different one for each platform.
2. Calculate how many different options you have for the entire campaign, assuming you can use the same banner for some or all the platforms.
3. Calculate how many ways we can pick which of the 8 banners to use, assuming we use different ones for each platform.
4. Calculate how many ways we can pick which of the 8 banners to use, assuming we can use each one multiple times.

Solutions: —

Now, we have 8 banners at our disposal, we need to put them on 5 platforms.

1. Using different banners for each platform means we can think of each social media platform as a different position. Hence, we are going to be dealing with Variations. Since, we are using different ones for each site/app, we cannot repeat values, so our formula is V = n! / (n-p)! => 8!/3! => 6,720.
2. Same as 1., but we can repeat values, so we have Variations with repetition, so v̅ = n^p => 8⁵ => 32,768.
3. We need to select 5 out of the 8 banners to use. We use different ones for each platform, so repetition is not allowed. Hence, we use the formula for Combinations without repetition. Thus, C = n! / p!(n-p)! => 8!/5!*3! => 56.
4. Now, we need to select 5 out of the 8 banners to use. However, we can choose some multiple times. Therefore, we need to use Combinations with repetition, so c = (n+p-1)!/p!(n-1) = 12!/5!*7! => 792.

PROBLEM 3

This year, you are helping organize your college’s career fest. There are 11 companies which are participating, and you have just enough room fit all of them. How many ways can you arrange the various firms, assuming: —

1. No firm has any preference where they want to be positioned?
2. JP Morgan representatives made a deal, where they have to be located exactly in the middle?
3. JP Morgan, Citi Bank and Morgan-Stanley must be positioned in the middle 3 spots?
4. Deutsche Bank representatives cancel, so you can give the additional space to one of the other companies?

Solutions: —

1. If no look has any preference, then we need to arrange the entire set of 11 firms across the room. Thus, we need to use Permutations, so: n! = 11! => 39,916,800.
2. If JP Morgan has to be located in the middle, then we only need to arrange the remaining 10 firms around the room. Thus, we can once again use Permutation, but this time n = 10. Therefore, n! = 10! => 3,628,800.
3. One approach to this problem is by looking at two distinct groups of firms — JP Morgan, Citi Bank, and MS as one group, and the other 8 firms as the second group. Then if we find the number of ways, we can set up each group around the room, we just have two events with distinct sample spaces.

• Let’s start with arranging the 3 banks in middle. Since we need to split the 3 middle spots among the 3 banks all we need to do is compute the number of Permutations among 3 elements. Therefore, 3! = 6.
• Now, since none of the remaining 8 firms cares too much where they are positioned, we once again rely on Permutations, so 8! = 40,320.
• For any of the 40,320 ways we set the 8 firms around the room, we have 6 different ways to arrange the 3 banks in the middle. Therefore, in total, we have 40,320 x 6 = 241,920 ways of setting up the career fest.

4. We have 10 firms, which need to fill out 11 spots. Then, if we start filling up the room in some specific order, then there are going to be 10 options for who gets the first position. Since, any firm can be given the additional space provided by DB’s withdrawal, then there are once again 10 options for the second spot. Then, there would be 9 different options for the third and so on. This results in having 10 x 10! = 36,288,000 many options to arrange the firms.