CSCE 235

Light Bulb 7

March 10, 2004

1. The important combinatorics are permutations, combinations, and repetitions. In the class, we also cover derangements and the binomial theorem. What is the most important application of combinatorics? Counting! It allows you to find out what the number of options is, what the number of ways is … and why is that important or useful? That is useful in probabilities and statistics. That allows you to know how a system behaves by finding the bounds. That allows you to plan. That allows you to design a solution to a problem with foresight. Areas such as decision theory in business administration, structural design in civil engineering, network management in computer science, route navigation in shipping, odds making in operational research, and so on, benefit from combinatorics.

2. Permutations are about putting distinct objects in an order. What’s so great about that? Seems like it is just about putting different-colored marbles into different boxes. Suppose you work for a travel agency. You are in charge of a tour that includes 10 tourist attractions in New Zealand. You want to plan a tour that visits each tourist attraction only once, in some order. How many possible ways are there? There are 10!. However, you also want to devise an economy tour package that only includes three tourist attractions, and once again, order is important. How many possible ways are there? There are ways. This gives you an upperbound on the number of possible designs that you can have. Now, you can apply your business sense to rule out tour routes that are not feasible (time consuming, logistics, etc.) and you can reduce the number of possible ways. Then you can present them to your boss for approval. This way, you know exactly how many designs there are. This way, you are confident that you have looked at all possible options.

3. Combinations are about putting non-distinct objects, regardless of order, into some boxes, with only one object to a box. This is an elegant way of solving many real world problems. For example, suppose you are a coach for a basketball camp. There are 30 kids in your camp. 4 of them are centers. 20 of them are forwards. And 6 of them are guards. Now, how many different teams can you form with 1 center, 2 forwards, and 2 guards? This is important since you want to have your assistants monitor each team to assess the kids’ performance in your camp. You also want to evaluate which five kids play best together as a team. So, now, you have the following number of different teams that you can form:

possible teams. Let say you have two teams play a 10-minute games. Thus, to see how the 11400 teams play, you would need (11400/2)*10 = 57000 minutes. That is 950 hours. If your camp lasts for 4 hours per day, how many days would it take to evaluate all teams? That would require 237.5 days!!! Now, you realize that would be a camp too long. Now, you can revise your plans. What about this? You now want a team to have 2 centers, 4 forwards, and 4 guards. How many different teams can you form?

Now, it is even worse. What should you do? May be you have set an unrealistic goal for your camp. Maybe you should simply form teams based on heights, weights, and age groups. That way, you introduce constraints into the design to obtain feasible results.

4. Repetitions are similar to combinations, but with greater freedom. It is a modeling of, say, user behavior, with greater freedom. It is about putting non-distinct objects, regardless of order, into some boxes, with any number of objects to a box. Instead of restricting users to choose one object only from a group, repetitions deal with users who may choose any number of objects from a group. For example, if you are given $10,000, for purchasing some computer equipments. There are computers, printers, and scanners. You may spend all of it on computers, or on computers and printers, and so on. If the $10,000 is divided into 20 $500 units, how many ways can you spend it on computer equipments? So, now we have 20 objects, 3 boxes: ==1540 ways.

Also for example, if you manage a charity and you are to give out $10,000 grants to students from five regions, and there are $500,000 in total grants. How many ways are there can you give out all $500,000? If you are required to give at least $25,000 total in grants to each region, then how many ways are there can you give out all $500,000?

The above examples show that you can plan better since you know better about the problem.

5. Derangements are about mistakes or out-of-order. If things do go wrong, how many ways can the worst scenarios occur? That is, if everything is out-of-order (the worst case), how many different ways can that occur? Why do we want to know that? Well, for quality assurance, for reliability assessment, for fault analysis, for testing, we sometimes want to know what are the ways that something could go wrong. And sometimes, we want to know what would happen if everything goes wrong. That way, we can plan for it, or avoid it, or design fail-proof methods, and so on. One way of doing this is to compute the derangement value. To find out the probability of such a derangement occurring. Then we will know how much effort we want to put into dealing with such a derangement.

For example, say you are the head of a package shipping company. You have 10 major customers that you daily ship packages to. What if one day, your computer breaks down and now the packages have been mis-addressed and sent to the wrong customers. What is the worst-case scenario? The worst-case scenario is when none of the customers receives its packages. We know the number of such scenarios is . That is our bottom line. Now, you may notice that some mistakes would never occur. For example, the packages for A will never be sent to B because of a prevention mechanism in your computer programs. Thus, you reduce the derangement value now. As you analyze your computer programs, you can try to lower the derangement value to a point that you can say that you are 80% resistant to worst-case scenarios; that is, the new derangement value is only 0.2 of the original derangement value. That allows you to measure the fault tolerance of your system against worst-case scenarios.

6. One fundamental importance of these combinatorics is for computing probabilities. Scientists, business administrators, doctors, engineers, and many other professionals deal with probabilities. Scientists use probabilities to compute the odds of an asteroid hitting Earth, to formulate theories about dinosaurs, to justify the possibility of a finding, etc. Business administrators use probabilities to decide which option is more viable, which action is more likely to yield more, etc. Doctors use probabilities in diagnosis, prognosis, etc. Engineers use probabilities to compute risks, to build fault tolerant systems, to analyze cost-effectiveness, etc. And what do combinatorics have to do with probabilities? Combinatorics allow us to find out the number of ways an event can occur. And that allows us to determine the probability of that event occurring. And that is a very useful, effective power of combinatorics.