The Importance of Permutations and Combinations in Modern Society

My friend, an English major, usually helps me edit my essays after I have written them. However, this time she refused, stating that she was “terrible at math” and might mess up my paper with her ignorance. Little did she know, we use permutations and combinations all the time without even realizing it and they are an integral part of modern life. In this essay I will discuss permutations and combinations, first defining these concepts, then showing examples, and relating them to practical applications.

A permutation is a way to arrange items or numbers if order matters. To know if order matters, think of this problem. If you are trying to visit three cities, does it matter which one you visit first? If it is a business trip, then the order probably matters because your boss wants you to spend the least amount of money possible. It is possible to fly certain routes more cheaply than others. However, if this were for your own vacation, perhaps you would not care in what order you visit these cities. But when the order matters, it is a permutation problem.

Here is an example of a permutation problem: pick three letters from ABCDE if ABC is not the same as CBA. So for the first letter, there are five choices. The choices are A, B, C, D, or E. For the second letter, there are only four choices since the same letter cannot be chosen twice. For the third letter, there are only three choices left to select from. Now multiply 5 x 4 x 3 to get the total number of permutations. So the answer is: when picking three letters from ABCDE, if ABC is not the same as CBA, there are sixty permutations. This problem can also be done quickly on a calculator, using the nPr button. The n stands for the number of items total, in this case five (ABCDE) and the r is the number that you want to pick, in this case three. So using 5P3 will also get sixty.

Another example of a permutation problem is “combination” locks. The name is actually misleading here, as this is not a combination problem, but rather a permutation problem. Order matters on a lock, as 123 is not the same as 132, 213, 231, 312, or 321. This problem cannot be worked out on a calculator using the nPr button however. So, for this problem, let us say that there are thirty-five possible numbers on the “combination” lock. The first time there are thirty-five numbers to select from, but the second time the lock can not be turned to the same number on the lock as the first, or no turning would take place. So for the second number, there are thirty-four numbers to select from. On the third number, it is possible to select the number that was picked the first time, but not the number that was picked the second time. So again there are thirty-four numbers left to pick from. Now multiply 35 x 34 x 34 to get the number of permutations possible with a thirty-five number “combination” lock. The answer would be that there are 40,460 permutations possible for this “combination” lock. What that essentially means is that if someone were trying to randomly guess your permutation, in one try they only have a one in 40,460 chance of guessing your permutation. That should make you feel fairly safe that your bike is safe from being stolen.

There are many applications of permutations in everyday life. One is a license plate. License plates are a permutation because the order matters. They are another one of the permutation problems that do not use the nPr button on a calculator. For example, there are seven spaces on a license plate. For the first space, it is possible to choose a number from one to seven. The next three spaces can have a letter from A to Z (and repeating is accepted), and the last three spaces can have a number from zero to nine (and repeating is also accepted). So that leaves seven possibilities for the first space, twenty-six for the second, third, and fourth, and ten for the fifth, sixth, and seventh. So that would be 7 x 26 x 26 x 26 x 10 x 10 x 10. There are 123,032,000 possible license plates not including personalized license plates. So if more than 123,032,000 cars are in California, then a new way of doing license plates will have to be decided.

Another application of permutations is telephone numbers. For example, if you dialed 123-4567 you’d get a different person than if you dialed 765-4321. For phone numbers, basically there are digits from zero to nine to choose from in each slot; however, there are a few rules. You cannot start with zero; that is for the operator. You cannot start with a one, since that is for the area code. Also, 911 is emergency, 611 is the telephone company, and 411 is information. So a phone number cannot start with a 911, 611, or a 411. Let us first figure out all the possible phone numbers, then subtract what you cannot have. That would be 10 x 10 x 10 x 10 x 10 x 10 x 10, or 107 possible phone numbers without restrictions.

That means there are 10,000,000 phone numbers that could be used, if not for certain rules. Now, find out what the restrictions will do to the numbers. If you want to find all the numbers with a one for the first digit, that leaves only one possible number for the first pick, and ten for the next six. That would be 1 x 10 x 10 x 10 x 10 x 10 x 10, which is 1,000,000. Therefore one million possible phone numbers cannot be used because of the one used for the area code. Another one million numbers cannot be used due to the zero used for the operator. Let us now find how many numbers cannot be used due to the other restrictions. To find how many numbers there would be with a 911 in front, take the first three digits as a given so that there is only one choice for each of those. There are ten choices for each of the other four digits. That would be 1 x 1 x 1 x 10 x 10 x 10 x 10. There are 10,000 numbers that cannot be used due to the 911 restriction.

There are another 10,000 numbers that cannot be used due to the 611 restriction and another 10,000 for the 411 restriction. That is a total of 2,030,000 numbers that cannot be used due to restrictions. Therefore, to find the total number of phone numbers for one area code, subtract 2,030,000 from the total 10,000,000. There are 7,970,000 phone numbers for each area code. If there are more telephones in the area then there are phone numbers, then a new area code is made and that is why you may get a new area code and number at some point in time. Eventually, we may also run out of area codes and then instead of seven digits all phone numbers might have eight digits.

A combination is a way to arrange items or numbers if order does not matter. Now you have to figure out if the order does not matter. If you must grab a yellow marble and a red marble, order might not matter. As long as you draw two marbles, you don’t need to pick the yellow first and the red second. Either way, you have both marbles. When order does not matter, it is a combination problem.

Here is an example of a combination problem: pick three letters from ABCDE if ABC is the same as CBA. This problem should look similar to the one mentioned earlier as a permutation problem. There are a few differences. First do the permutation part of the problem. However, there are ways included in that problem that are the same. For example, ABC, ACB, BAC, BCA, CAB, and CBA are all considered the same in this problem. Therefore, divide your answer from the permutation problem by the number of choices that are the same to get the number of combinations possible. The number of choices that are the same for each three letters chosen are six. To choose three, multiply 3 x 2 x 1 (also known as 3! or three factorial) to get the six. Sixty divided by six would be ten. There are ten combinations when choosing three letters from ABCDE. This is also possible to do on a calculator using the nCr button. Let n equal the number of items total and r equal the number of items that need to be chosen. Therefore 5C3 would also equal ten. The number of combinations is always smaller than the number of permutations for the same problem.

Another example of a combination is the lottery. In the California Lotto, it is not required that you pick the numbers in the order that they appear. The available numbers are from zero through fifty and you must choose six numbers. Since this is not a permutation problem you probably think you have a great chance of winning now. Well, not exactly. Let us find your chances of winning the California lotto. There are fifty-one numbers to choose from on the first number, and then that number cannot be drawn out again because the numbers are not replaced. On the second number there are fifty choices, the third forty-nine, the fourth forty-eight, the fifth forty-seven, and the sixth forty-six. Multiply 51 x 50 x 49 x 48 x 47 x 46. Then since it is a combination problem, divide by 6 x 5 x 4 x 3 x 2 x 1 (or 6!) to get rid of the choices that are the same. This can also be done with the nCr button on the calculator, as 51C6. Your answer is 18,009,460. Therefore, if you only buy one lottery ticket, your chances of winning are only one in about 18 million.

There are also many applications for combinations. One is ordering a pizza. When ordering a pizza, it really doesn’t matter what order the toppings are put onto the pizza. So if there are eight toppings to choose from, and you want two toppings, that is a combination problem. There have eight toppings to choose from for the first choice, then seven for the second choice. That would be 8 x 7, divided by two, which is twenty-eight. There are twenty-eight combinations of ordering two toppings from a list of eight. Let’s say you need an outfit to wear to this pizza place. You are short on cash, and do not want to buy many of clothes. You want a different outfit every day of the month however. You bought two pairs of shoes, five pairs of pants, and six shirts. Multiply 2 x 5 x 6 and then there are sixty possible outfits, about twice the amount needed to have a different outfit everyday of the month.

In modern life we need order, even in a society of chaos and this is just one more way to put things in order. So now you know what permutations and combinations are, some examples, and how to use them. No matter where you are, no matter what you do – even if you are an English major – you still need and use combinations and permutations.