Permutations and combinations are the next step in the learning of probability. It is by using permutations and combinations that we can find the probabilities of various events occurring at the same time, such as choosing 3 boys and 3 girls from a class of grade 12 math students.
In mathematics, we use more precise language:
If the order doesn't matter, it is a combination.
If the order does matter, it is a permutation.
Say, for example, you are making a salad. You throw in some lettuce, carrots, cucumbers, and green peppers. The order in which you throw in these vegetables doesn't really matter. Here we are talking about a combination. For combinations, you are merely selecting. Say, though, that Jack went to the ATM to get out some money and that he has to put in his PIN number. Here the order of the digits in the PIN number is quite important. In this case, we are talking about a permutation. For permutations, you are ordering objects in a specific manner.
How many ways can you arrange the letters in the word MATH?
You have 4 letters in the word, and you are going to choose 1 letter at a time. When you choose the first letter, you have 4 possibilities ('M', 'A', 'T', or 'H'). Your second choice will have 3 possibilities, your third choice will have 2 possibilities, and your last choice will have only 1 possibility.
The factorial function (!) requires us to multiply a series of descending natural numbers.
With TI calculators, you can find the factorial function using:
This represents the number of ways to arrange 4 objects that are chosen from a set of 4 different objects.
Starting with 6, multiply the first 3 numbers of the factorial:
This represents the number of ways to arrange 3 objects that are chosen from a set of 6 different objects.
The formula to solve permutations like these is:
What is the total number of possible 4-letter arrangements of the letters 's', 'n', 'o', and 'w' if each letter is used only once in each arrangement?
A committee is to be formed with a president, a vice president, and a treasurer. If there are 10 people to select from, how many committees are possible?
Permutations with Repetition
There is a subset of permutations that takes into account that there are double objects or repetitions in a permutation problem. In general, repetitions are taken care of by dividing the permutation by the factorial of the number of objects that are identical.
If you look at the word TOOTH, there are 2 O’s in the word. Both O’s are identical, and it does not matter in which order we write these 2 O’s, since they are the same. In other words, if we exchange 'O' for 'O', we still spell TOOTH. The same is true for the T’s, since there are 2 T’s in the word TOOTH as well.
If we were to ask the question, "In how many ways can we arrange the letters in the word TOOTH?" we must account for the fact that these 2 O’s are identical and that the 2 T’s are identical. We do this using the formula:
Tech Tip: Calculating Permutations on the Calculator
How many different 5-letter arrangements can be formed from the word APPLE?
How many different 6-digit numerals can be written using the following 7 digits? Assume the repeated digits are all used.
3, 3, 4, 4, 4, 5, 6
When there are no repetitions, remember that we use the standard permutation formula:
In how many ways can first and second place be awarded to 10 people?
In how many ways can 3 favorite desserts be listed in order from a menu of 10 (i.e., permutations without repetition)?
If you think about the lottery, you choose a group of lucky numbers in hopes of winning millions of dollars. When the numbers are drawn, the order in which they are drawn does not have to be the same order as on your lottery ticket. The numbers drawn simply have to be on your lottery ticket in order for you to win. You can imagine how many possible combinations of numbers exist, which is why your odds of winning are so small!
In how many ways can 3 desserts be chosen in any order from a menu of 10?
There are 12 boys and 14 girls in Mrs. Cameron's math class. Find the number of ways Mrs. Cameron can select a team of 3 students from the class to work on a group project. The team must consist of 2 girls and 1 boy.
Therefore, the number of ways Mrs. Cameron can select a team of 3 students (2 girls and 1 boy) from the class of 26 students to work on a group project is:
If there are 20 rock songs and 20 rap songs to choose from, in how many different ways can you select 12 rock songs and 7 rap songs for a mix CD?
As in Example 15, we have multiple groups from which we are required to select, so we have to calculate the possible combinations for each group (rock songs and rap songs in this example) separately and then multiply together.
Therefore, the total number of possible combinations is:
Tech Tip: Calculating Combinations on the Calculator