N Over R Calculator

The n over r calculator helps you compute combinations and permutations quickly. Whether you're working on probability problems, statistical analysis, or combinatorial mathematics, this tool provides accurate results for the n over r calculation.

What is n over r?

In combinatorics, "n over r" refers to the number of ways to choose r items from a set of n distinct items without regard to the order of selection. This is commonly known as combinations and is represented mathematically as C(n, r) or "nCr".

The calculation is fundamental in probability, statistics, and discrete mathematics. It helps determine the number of possible outcomes in scenarios where order doesn't matter, such as lottery combinations, poker hands, or selecting teams from a group.

How to calculate n over r

Calculating n over r involves a straightforward mathematical process. Here's a step-by-step guide:

Identify the total number of items (n)
Determine how many items you want to choose (r)
Calculate the factorial of n (n!)
Calculate the factorial of r (r!)
Calculate the factorial of (n - r) ((n - r)!)
Divide n! by the product of r! and (n - r)!

The result is the number of combinations of n items taken r at a time.

n over r formula

The formula for n over r is:

C(n, r) = n! / (r! × (n - r)!)

Where:

n! = n × (n - 1) × (n - 2) × ... × 1
r! = r × (r - 1) × (r - 2) × ... × 1
(n - r)! = (n - r) × (n - r - 1) × ... × 1

This formula accounts for all possible combinations by considering the unique groupings of r items from a set of n items.

n over r examples

Let's look at some practical examples to understand how n over r works:

Example 1: Selecting a committee

Suppose you have 10 people and want to form a committee of 3 members. The number of possible committees is calculated as:

C(10, 3) = 10! / (3! × 7!) = 120

This means there are 120 different ways to choose 3 people from 10.

Example 2: Lottery combinations

In a lottery where you pick 6 numbers from 49, the number of possible combinations is:

C(49, 6) = 49! / (6! × 43!) = 13,983,816

This shows the vast number of possible outcomes in such games.

Example 3: Poker hands

When dealing a 5-card hand from a standard 52-card deck, the number of possible hands is:

C(52, 5) = 52! / (5! × 47!) = 2,598,960

This calculation is essential for understanding probability in card games.

n over r table

Here's a reference table showing n over r values for small numbers:

n	r	n over r (C(n, r))
5	2	10
6	3	20
7	4	35
8	5	56
9	6	84
10	7	120

This table provides quick reference points for common n over r calculations.

FAQ

What is the difference between combinations and permutations?

Combinations (n over r) count the number of ways to choose items without considering order, while permutations count the number of ways to arrange items where order matters. For example, choosing a committee of 3 from 5 people is a combination, while arranging 3 people in a line is a permutation.

When should I use the n over r calculator?

Use the n over r calculator whenever you need to determine the number of possible combinations in scenarios like selecting teams, lottery numbers, poker hands, or any situation where order doesn't matter. It's particularly useful in probability and statistics.

Can n over r be greater than n?

No, n over r cannot be greater than n. The maximum value occurs when r = n/2 (rounded down), and the value decreases symmetrically as r moves away from this point. For example, C(5, 2) = 10, which is less than 5.

What happens when r is 0 or equal to n?

When r is 0, n over r is always 1 because there's exactly one way to choose nothing from a set. When r equals n, n over r is also 1 because there's only one way to choose all items from the set.