Combination Formula: Definition, Uses in Probability, Examples & More

The combination formula is used to calculate the number of ways of selecting events from a collection of events, such that the order of selection does not matter. In simple words, combination involves the selection of events out of a larger group where order doesn’t matter.
To calculate the probability of a combination, you will need to consider the number of favorable events over the number of total events. Combinations are used to calculate events where the order of events does not matter. In this article, we will explore the connection between these two essential topics.

Note: The formulas in this article assume that we have no substitution, which implies that items cannot be repeated.

Combination Formula
Combination Formula

Combinations are a method to calculate the total events of an event where the order of the events does not matter. To calculate combinations, we will use the combinations formula

nCr = n! / r! * (n – r)!,

where n stands for the number of items, and r stands for the number of items being chosen at a time.

Combinations Formula

Looking at the equation to calculate combinations, you can see that factorials are used throughout the formula. Remember, the formula to calculate combinations is

nCr = n! / r! * (n – r)!,

where n stands for the number of items, and r stands for the number of items being chosen at a time.

Notations in combinations Formula

  • r is the size of each permutation
  • n is the size of the set from which elements are permuted
  • n, r are non-negative integers
  • ! is the factorial operator

Factorial

To calculate a combination, you will need to find out a factorial. A factorial is the multiplication of all the positive integers equal to and less than your number. A factorial is written as the number with an exclamation point.

For example, to write the factorial of 4, we would write 4!. To find out the factorial of 4, you would multiply all of the positive integers equal to and less than 4. So, 4! = 4 * 3 * 2 * 1. By multiplying these numbers, we can find that 4! = 24.

Let’s look at other examples, to write the factorial of 6, we would write 6!. To calculate 6!, we would multiply 6 * 5 * 4 * 3 * 2 * 1, and that equals 720.

How to calculate a combination

There are ten new movies out to rent this week on DVD. Rahul wants to select three movies to watch this weekend. How many combinations of movies can he select?

In this problem, Rahul is choosing three movies from the ten new releases. 10 would represent the n variable, and 3 would represent the r variable. So, our equation would look like

10C3 = 10! / 3! * (10 – 3)!.

The first step is to be done is to subtract 3 from 10 on the bottom of this equation.

10 – 3 = 7,

so our equation becomes 10! / 3! * 7!.

Next, we need to expand each of our factorials.

10! would equal 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 on the top,

and 3! * 7! would be 3 * 2 * 1 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

The finest way to work this example is to cancel out like terms. We can see that there is a 7, 6, 5, 4, 3, 2, and 1 on both the numerator and denominator of our equation. These numbers can be canceled out.

We now see that our equation has 10 * 9 * 8 left on top and 3 * 2 * 1 left on the denominator. From here, we can just multiply. 10 * 9 * 8 = 720, and 3 * 2 * 1 = 6. So, our equation is now 720 / 6.

To finish this example, we will divide 720 by 6, and we get 120. Rahul now knows that he could select 120 different combinations of new-release movies this week.

Probability of Combinations

To find out the probability of an event happening, we will use the formula:

The number of favorable events / the number of total events

Let’s look at an example of how to find out the probability of an event appearing. At the checkout in the DVD store, Rahul also purchased a bag of gumballs. In the bag of gumballs, there were five red, three green, four white, and eight yellow gumballs. What is the probability that Rahul drawing at random will select a yellow gumball?

Rahul knows that if he adds all the gumballs together, there are 20 gumballs in the bag. So, the number of total events is 20. Rahul also knows that there are eight yellow gumballs, which would represent the number of favorable events. So, the probability of selecting a yellow gumball at random from the bag is 8 out of 20.

All fractions, however, must be simplified. So, both 8 and 20 will divide by 4. So, 8/20 would reduce to 2/5. Rahul knows that the probability of him selecting a yellow gumball from the bag at random is 2/5.

To calculate the number of total events and favorable events, you might have to calculate a combination. Remember, a combination is a method to find out events where the order events do not matter.

Let’s look at an example.

Example 1

A 4 digit PIN is selected. What is the probability that there are no repeated digits?

There are 10 possible values for each digit of the PIN (namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), so there are 10 × 10 × 10 × 10 = 10
10P4 = 10000 total possible PINs.

To have no repeated digits, all four digits would have to be different, which is selecting without replacement. We could either compute 10 × 9 × 8 × 7 or notice that this is the same as the permutation
10P4 = 5040.

The probability of no repeated digits is the number of 4 digit PINs with no repeated digits divided by the total number of 4 digit PINs. This probability is

10P410P4104 = 504010000 = 0.504

Example 2

In a certain state’s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers are drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in doesn’t matter. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket.

In order to compute the probability, we need to count the total number of ways six numbers can be drawn, and the number of ways the six numbers on the player’s ticket could match the six numbers drawn from the machine. Since there is no stipulation that the numbers be in any particular order, the number of possible outcomes of the lottery drawing is
48C6 = 12,271,512. Of these possible outcomes, only one would match all six numbers on the player’s ticket, so the probability of winning the grand prize is:

Combinations Formula Calculator

The Combinations Calculator will find the number of possible combinations that can be obtained by taking a sample of events from a larger set. Basically, it shows how many different possible subgroups can be made from the larger set. For this calculator, the order of the items chosen in the subset does not matter.

Combinations Calculator
Combinations Calculator

FAQ’s

What Is An Example Of A Combination?
A Combination Is A Selection Of All Or Part Of A Set Of Objects, Without Regard To The Order In Which Objects Are Selected. For Example, Suppose We Have A Set Of Three Letters: A, B, And C. Each Possible Selection Would Be An Example Of A Combination.
How Many Combinations Of 5 Items Are There?
Hence There Are 2 Choices For The Fourth Digit And Only 1 Choice For The Fifth Digit. Thus You Have Made 5 × 4 × 3 × 2 1 = 120 Choices And There Are 120 Possible 5 Digit Numbers Made From 1, 2, 3, 4 And 5 If You Don’t Allow Any Digit To Be Repeated. Now Consider The Possibilities With 13 As The First Two Digits.
How Many Combinations Of 4 Items Are There?
You Multiply These Choices Together To Get Your Result: 4 X 3 X 2 X 1 = 24.
How Many 10 Digit Combinations Are There?
If Repetition Is Allowed, Then The Number Of Permutations Of 10 Digits Is 10,000,000,000.
How Do You Calculate The Number Of Possible Combinations?
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.
What Are All The Possible Combinations Of 1234?
If You Wager On 1234 Boxed, You Would Win If Any Of The Following Combinations Were Drawn: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, Or 4321.