High School Math Help: Combinations and Permutations

Permutations and Combinations

High school students learn to calculate the number of possible combinations or permutations in a given situation, and then find the probability that a particular one will occur. Here's a sample problem:

For an upcoming field trip, Amanda's teacher is dividing her 20-student class into groups of four students. What is the probability that Amanda will be in a group with her three best friends, Sammy, Leslie and Jamie?

The probability that these four students will be in a group together is one out of the total number of combinations the teacher could choose. Keep reading to learn formulas for calculating combinations and permutations in different scenarios!

Permutation Formulas

When you calculate the number of ways you can arrange a set of things, each possible arrangement is called a permutation. For instance, each of the 5-digit lottery numbers you can draw from the digits one through five is a permutation. A permutation without repetition means you can only use each digit once per permutation.

The formula for this is n!, where n is the number of things you're arranging and n! = (n) x (n - 1) x (n - 2) x … x (1). If n = 5, then n! = 5 x 4 x 3 x 2 x 1 = 120. This means that 120 different lottery numbers are possible.

You'll also need to calculate permutations without repetition when you're choosing a few things out of a larger group. For instance, to calculate how many 3-digit lottery numbers you can draw from the numbers one through five, use the formula n!/(n - r)!, where n is the total number of possibilities (five) and r is the number of items in each permutation (three).

Sometimes, permutations are allowed to contain repeating digits. For instance, your locker number could be 11111 or 99888. This is called a permutation with repetition. The formula to calculate this is n^r, where n is the number of choices for each digit, and r is the number of digits in each permutation.

Combination Formulas

With permutations, we count the number of ways that things can be arranged. With combinations, we only count the number of possible groupings. Going back to the example with Amanda, there are lots of different ways we could arrange her and her friends if they were standing in a line. However, we can make just one combination with these four students, because we're only concerned with who's in the group, not the physical order in which they're arranged.

To calculate the number of 4-student combinations we can make out of Amanda's entire 20-student class, we need to use the formula for combinations without repetition: n!/r!(n - r)!. In this example, n = 20 and r = 4. There is no repetition because we can't have the same person in a group twice.

Sometimes, you can have repetition within combinations. For instance, if you're choosing a combination of three pizza toppings out of a list of ten, you could choose pepperoni, olives and mushrooms or just a triple order of pepperoni. To calculate the number of possible combinations with repetition, use this formula: (n + r - 1)!/r!(n - 1)!.