Combination Calculator

How to Use the Combination Calculator

This combination calculator allows you to find the number of ways you can choose r items from n items, when the order does not matter. If the order matters, use the permutation calculator instead.

To use the calculator:

1. Enter the total number of available items, n.
2. Enter the number of items you want to choose, r.
3. Leave the With repetition toggle button turned off for a standard combination. However, if the problem says that the same item can be selected more than once, turn on the With repetition toggle.
4. Click Calculate.

The calculator instantly returns the correct combination solution and shows how the value was calculated manually, step-by-step.

What Is a Combination?

A combination is a selection of items in which the order of selection does not matter. For example, suppose you need to choose 2 students from Ava, Ben, and Chloe. Choosing Ava and Ben gives the same group as choosing Ben and Ava. Since changing the order does not create a new group, this is a combination.

In a combination:

• n is the total number of available items.
• r is the number of items being chosen.
• Order does not matter.

For example, if you choose 6 students from a group of 10:

• n = 10
• r = 6

The number of possible groups is written as C(10, 6), 10C6, or “10 choose 6.”

When Should You Use Combinations?

You should use combinations when you want to select a group or a collection of items where the order of selection does not matter.

Common examples include:

• choosing students for a committee;
• selecting cards from a deck;
• choosing lottery numbers;
• selecting questions from an exam question bank;
• forming teams from a group of people; or
• selecting several products from a larger set.

The easiest way to determine whether a problem involves combinations is to ask:

Would changing the order of the selected items create a different outcome? If the answer is no, use a combination.

For example, selecting Maya, James, and Noah for a committee is the same committee regardless of the order in which their names were selected. If order does matter, the problem involves a permutation rather than a combination.

Combination Formula

The standard formula for finding combinations is:

Where:

• n is the total number of available items;
• r is the number of items being chosen; and
• ! represents factorial.

The above formula only works when order does not matter, and repetition is not allowed. However, if repetition is allowed, the combination formula becomes:

Note. For the combination formula to work, 0 ≤ r ≤ n

What Does Factorial Mean?

A factorial is the product of a positive whole number and all the positive whole numbers below it.

For example: 5! = 5 × 4 × 3 × 2 × 1

= 120

Similarly, 3! = 3 × 2 × 1

= 6

Want to quickly find factorials when working with combination problems? Use the factorial calculator.

How to Calculate Combinations

To calculate a combination manually:

1. Identify the total number of available items (n).
2. Identify the number of items being chosen (r).
3. Apply the correct formula and solve

Example 1: Combination Without Repetition

A college has 12 students who have volunteered for a debate team. Four students will be selected to represent the college at a competition. How many different groups of 4 students can be selected?

Solution

To solve the problem manually, follow these steps:

Step 1: Identify the total number of available items (n).

There are 12 students available for selection.

Therefore, n = 12

Step 2: Identify the number of items being chosen (r).

Four students will be selected. Thus, r = 4

Step 3: Apply the correct formula and solve

From the question, a student cannot be selected more than once, and the order in which the four students are selected does not matter. Therefore, this is a combination without repetition, so we should use the standard combination formula:

Recall. The standard combination formula is: ⁿC_r = n!/r!(n-r)!

Substituting n = 12 and r = 4, we get:

¹²C₄ = 12!/4! (12-4)!

= 12!/4! x 8!!

Evaluating each factorial, we get:

¹²C₄ = 479,001,600/(24 x 40,320)

= 479,001,600 / 967,680

= 495

This implies that 495 different groups of 4 students can be selected from the 12 students. In other words, ¹²C₄ = 495.

Example 2: Combination With Repetition

An ice cream shop offers 6 flavors. A customer wants to choose 4 scoops, and the same flavor can be selected more than once. How many different flavor combinations are possible if the order of the scoops does not matter?

Solution

To solve the above problem by hand, follow these steps:

Step 1: Identify the total number of available items (n).

There are 6 available ice cream flavors. Therefore, n = 6

Step 2: Identify the number of items being chosen (r).

The customer is choosing 4 scoops. Therefore, r = 4

Step 3: Apply the correct formula

Since the same ice cream flavor can be selected more than once, and the order of the scoops does not matter, this is a combination problem with repetition.

By definition, the combination formula that accounts for repetition is: ^n+r-1C_r= (n+r-1)!/r! (n-1)!

Substituting the values into the formula, we get:

^6+4-1C₄= (6+4-1)!/4!(6-1)!

Thus, ⁹C₄ = 9!/4!5!

Evaluating each factorial, we get:

⁹C₄ = 362,880/(24 x 120)

= 362,880/2,880

= 126

Therefore, there are 126 different combinations of 4 scoops that can be selected from the 6 flavors when repetition is allowed.

Why Does Order Not Matter in a Combination?

Suppose you choose 2 letters from A, B, and C. The possible ordered selections include: AB, BA, AC, CA, BC, CB

However, in a combination:

• AB and BA represent the same selection.
• AC and CA represent the same selection.
• BC and CB represent the same selection.

Therefore, there are only 3 unique combinations: AB, AC, BC

The combination formula removes these duplicate arrangements automatically.

Useful Combination Rules

The following combinatorial properties can make combination calculations easier to understand.

• Choosing zero items. There is exactly one way to choose nothing from a set (i.e., ⁿC₀ = 1)
• Choosing every item. There is exactly one way to choose all available items (i.e., ⁿC_n = 1)
• Choosing one item from n available items always gives you n different possibilities (i.e., ⁿC₁ = n)
• Symmetry of combinations states that choosing a group of items is identical to choosing which items to leave behind. (i.e., ⁿC_r = ⁿC_{n − r})

How Are Combinations Used in Probability?

Combinations are frequently used in probability when we need to count the number of possible groups of outcomes.

For example, the probability of obtaining exactly x successes in a binomial experiment contains the combination: C(n, x). This counts the number of different ways x successes can occur among n trials.

For example, if 3 successes occur in 5 trials, the number of possible arrangements of those successes is C(5, 3) = 10. This is why combinations appear in the binomial probability formula. Therefore, when working on a binomial probability problem, you may find this combination calculator useful.

Common Mistakes When Calculating Combinations

Avoid these common mistakes:

• Using a combination when order matters. If changing the order creates a different outcome, you need a permutation.
• Confusing n and r. n is the total number of available items, while r is the number being selected.
• Entering r greater than n for a standard combination. Without repetition, you cannot choose more distinct items than are available.
• Forgetting whether repetition is allowed. Standard nCr assumes that each item can be selected only once.
• Counting different orders as different combinations. For combinations, selecting A, B, C is the same as selecting C, A, B.
• Forgetting that 0! = 1. This is especially important when calculating C(n, n) or C(n, 0).
• Using the standard combination formula for a problem with repetition. If items can be selected repeatedly, turn on the With repetition? option and use the repetition formula.

Frequently Asked Questions