What happens when ABC and CBA are the same thing? Learn how to describe and calculate combinations. Explore differences between permutations and combinations in this video.
- When you're giving out three cash prizes in a raffle, $10, $100, $1,000, the order in which the three winners are chosen is important. In those cases, when order is important, we use permutations, but sometimes order doesn't matter. In those cases, we use combinations. Consider this scenario. Suppose we have 12 students in a class and one team of four students is going to be chosen at random to represent the school at a prestigious competition. Olivia and Layla are sisters. They both want to be on the team together, so what's the probability they will both get chosen for the team? To solve this, we'll need to know how many combinations of four students would be possible. Combinations are different than permutations. How? Well, look at these two teams of students. In both cases, we have the same combination of students. When we create teams, the order in which we list the names doesn't change the nature of the team. So both of these teams actually represent the same combination of students. Now that we understand what a combination is, here is the formula used to find the total number of combinations. N factorial divided by the quantity N minus X factorial times X factorial, where N is the total number of objects and X is the number of objects chosen at one time. In this case, we have 12 students in the class, so that is N. One team of four will be chosen, so X is equal to four. 12 factorial divided by the quantity 12 minus four factorial times four factorial. If we cancel the redundant items on both top and bottom, we get this, which tells us that there are 495 possible teams of four, but how many of those 495 combinations have both Layla and Olivia? Well, think of it this way. Olivia and Layla take up two spots, and we now have two spots open for the other 10 students. To solve this, we need to figure out how many combinations of our 10 remaining students can fill those last two spots. In our formula, 10 is the total number of students. X is two, the number of open slots. 10 factorial divided by the quantity 10 minus two factorial times two factorial. Once we cancel out the redundant items, we find this gives us 45. So remember, there are 495 total combinations of four students in a class of 12 students. And we just found out that there are 45 combinations where Layla and Olivia end up on the same team. 45 desired outcomes divided by 495 total outcomes. There is a 9.1% chance that Olivia and Layla will both be on the school's team. As you can see, being able to calculate combinations is a very important skill for those that are interested in probabilities.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.