How To Calculate The Number Of Combinations

A "combination" is an unordered series of distinct elements. An ordered series of distinct elements is referred to as a "permutation." A salad may contain lettuce, tomatoes and olives. It does not matter what order it is in; you can say lettuce, olives and tomatoes, or olives, lettuce and tomatoes. In the end, it's still the same salad. This is a combination. The combination to a padlock, however, must be exact. If the combination is 40-30-13, then 30-40-13 will not open the lock. This is known as a "permutation."

Step 1

Review combination notation. Mathematicians use nCr to notate a combination. The notation stands for the number of "n" elements, taken "r" at a time. The notation 5C3 indicates the number of combinations in which 3 elements can be selected out of 5.

Step 2

Review factorials. Mathematicians use factorials to solve combination problems. A factorial represents the product of all numbers from 1 up to (and including) the specified number. Thus, 5 factorial = 12345. "5!" is the notation for "5 factorial."

Step 3

Define the variables. To best understand the concept, let's work through an example. Let's look at the number of ways 13 playing cards can be selected from a deck of 52. The first card selected can be any one of the 52 cards. The second number selected is taken from 51 cards and so on.

Step 4

Review the formula for combinations. The formula for combinations is generally n! / (r! (n — r)!), where n is the total number of possibilities to start and r is the number of selections made. In our example, we have 52 cards; therefore, n = 52. We want to select 13 cards, so r = 13.

Step 5

Substitute the variables into the formula. To know how many combinations of 13 can be selected from a deck of 52 cards, the equation is 52! / 39! (13!) or 635,013,559,600 different combinations.

Step 6

Check your calculation with an online calculator. Use the online calculator found in Resources to validate your answer.