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How to Do C N K on A Calculator

Reviewed by Calculator Editorial Team

Combinations (C(n, k)) are a fundamental concept in combinatorics that calculate the number of ways to choose k items from n items without regard to order. This guide explains how to calculate C(n, k) using a calculator, including manual methods and practical examples.

What is C(n, k)?

C(n, k) represents the number of combinations of n items taken k at a time. It's calculated using the formula:

Combination Formula

C(n, k) = n! / (k! × (n - k)!)

Where "!" denotes factorial, the product of all positive integers up to that number.

Combinations are used in probability, statistics, and many real-world applications like lottery odds, committee selection, and genetic probability calculations.

Using a Calculator

Most scientific calculators have a built-in combination function. Here's how to use it:

  1. Enter the value of n (total items)
  2. Press the combination function button (often labeled "nCr" or "C")
  3. Enter the value of k (items to choose)
  4. Press "=" to get the result

Note

If your calculator doesn't have a combination function, you can calculate it manually using factorials as shown in the next section.

Manual Calculation

When you don't have a calculator with combination functions, you can calculate C(n, k) using factorials:

  1. Calculate n! (n factorial)
  2. Calculate k! (k factorial)
  3. Calculate (n - k)! ((n - k) factorial)
  4. Multiply k! and (n - k)! together
  5. Divide n! by the product from step 4

Factorial Calculation

n! = n × (n - 1) × (n - 2) × ... × 1

For example, 5! = 5 × 4 × 3 × 2 × 1 = 120

Worked Example

Let's calculate C(5, 2) (the number of ways to choose 2 items from 5):

  1. Calculate 5! = 120
  2. Calculate 2! = 2
  3. Calculate (5 - 2)! = 3! = 6
  4. Multiply 2! × 3! = 2 × 6 = 12
  5. Divide 120 / 12 = 10

The result is 10, meaning there are 10 different ways to choose 2 items from 5.

FAQ

What's the difference between combinations and permutations?

Combinations (C(n, k)) don't consider order, while permutations (P(n, k)) do. For example, choosing apples and oranges is the same as choosing oranges and apples in combinations, but different in permutations.

When would I use combinations in real life?

Combinations are useful for calculating probabilities, lottery odds, committee selections, and any situation where order doesn't matter.

What if n is less than k?

If n is less than k, the result is 0 because you can't choose more items than you have. For example, C(3, 5) = 0.