Permutation and Combination Calculator

What Are Permutations and Combinations?

Given a sample size, a permutation is the number of ways a certain number of objects can be arranged in order. A combination, on the other hand, is the number of ways a certain number of items can be grouped together for a given sample size. Both are useful when calculating probabilities.

For example, what is the probability of winning the lottery? To win, you need to get the right numbers in the right order. So to answer that question, you need to understand permutations. A similar idea applies to combinations.

How to Calculate Permutations and Combinations?

Permutation example:

1. Determine the total number of objects n (the total items you have; in this example n = 6).
2. Determine the sample size r (the size of the permutation you want; in this example r = 3).
3. Calculate the number of possible permutations using the formula nPr = n! / (n-r)!; the number of permutations is nPr = 6! / (6-3)! = 120.

Combination example:

1. Determine the total number of objects n (in this example n = 7).
2. Determine the sample size r (in this example r = 4).
3. Calculate the number of possible combinations using the formula nCr = n! / (r!(n-r)!); the number of combinations is nCr = 7! / (4! × (7-4)!) = 35.

What Is the Difference Between Permutations and Combinations?

• Permutations: count the possible ways to arrange a set of elements; different sequences of the same elements are considered different permutations. For example, ABC and BCA are two different permutations.
• Combinations: sequences containing the same elements are considered the same; ABC and BCA are the same combination.
• Permutations and combinations solve different probability problems: permutations are used for sequence problems like the lottery, while combinations are mainly used for problems that ignore order.