Combinations Calculator

How to Calculate Combinations

Combinations are a way of selecting items from a collection, such that the order of selection does not matter. In other words, it's the number of ways to choose r items from a set of n items, where the order doesn't matter.

Combinations Formula

The formula for calculating combinations is:

\[ C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where:

• C(n,r) is the number of combinations
• n is the total number of items to choose from
• r is the number of items being chosen
• ! denotes factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)

Calculation Steps

1. Identify the values of n and r
2. Calculate n!
3. Calculate r!
4. Calculate (n-r)!
5. Apply the formula: C(n,r) = n! / (r! * (n-r)!)

Example Calculation

Let's calculate the number of combinations for choosing 3 items from a set of 5 items.

1. n = 5, r = 3
2. 5! = 5 × 4 × 3 × 2 × 1 = 120
3. 3! = 3 × 2 × 1 = 6
4. (5-3)! = 2! = 2 × 1 = 2
5. C(5,3) = 120 / (6 × 2) = 10

Therefore, there are 10 ways to choose 3 items from a set of 5 items.

Visual Representation

Here's a visual representation of combinations:

This diagram illustrates choosing 3 items (red) from a set of 5 items. The number of different ways to make this selection is the number of combinations, C(5,3) = 10.