How to Calculate Combinations of Two Groups of Numbers

Combinations are the ways in which items can be grouped.

Combinations are all around us, and those people who understand them can be at an advantage over those who do not. From calculating the chances of getting a particular hand in a game of cards through to planning a school football tournament, the math behind the planning is all about combinations. Calculating the total number of ways to combine the numbers in two different groups is a straightforward process for anyone with access to a scientific calculator.

Things You'll Need

• Calculator

Instructions

1. Establish how many numbers, or elements, there are in each group. The values of the individual numbers is not important, just the total number of elements in each group.

   For example, if a group contains 1, 7, 3 and 22, there are four elements in the group. Add the totals for both groups to establish the overall number of elements. This value is known as "n."

2. Determine "r," the size of the combinations. For example, any number of elements combined into groups of three has an "r" value of three.

3. A factorial of a number is the value of the number multiplied by every whole number smaller than itself down to one, so 4! is the same as 4x3x2x1. The "!" sign means factorial.

   Substitute the values for "n" and "r" into the formula: C = n! / r!(n-r)! where C stands for the number of possible combinations. For example, with n = 10 and r = 3, the formula becomes C = 10! / 3!(10-3)!

4. Use the factorial button on a calculator, or long multiplication, to determine the value of each factorial in the equation. Using the example above C = 3628800 / 6 x 5040. The result, in this example 120, is the number of possible combinations of two groups of "n" numbers, in sets of size "r."

Tips & Warnings

• To find the number of combinations of pairs, with one element from each group, multiply the number of elements in one group by the number of elements in the other group. For example, with groups of 10 and 12 numbers, there are 120 possible pairs.

• Combinations do not consider the order of the elements, so AB is the same as BA. Use Permutations if the order of the elements is important.

• Factorials rapidly become very large numbers. The factorial of 100 is roughly 9.3 with more 150 zeros after it!

References

• Buffalo State College; Permutations and Combinations; Dan Tracz; July 2004
• Drexel University - Ask Dr. Math: Permutations and Combinations
• WolframMathWorld; Factorial; Eric Weisstein
• QuickMBA: Permutations and Combinations
• Georgia State University - HyperPhysics; Permutations and Combinations; C.R. Nave; 2005