How to Calculate the Number of Combinations

Calculate 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."

    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.

    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 = 1_2_3_4_5. "5!" is the notation for "5 factorial."

    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.

    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.

    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.

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


    • You can also calculate combinations in Excel using the function COMBIN. The exact formula is: =COMBIN(universe, sets). The number of four-character combinations that can be made from the alphabet is: =COMBIN(26, 4) or 14,950.

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