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Combinations (nCr) Calculator

Calculates the number of ways to choose 'r' items from a set of 'n' items without regard to the order of selection.

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No direct algebraic solution for $n$ in $\binom{n}{r} = C$. Requires numerical methods or specific algebraic simplification for small $r$ values, as there is no general closed-form algebraic rearrangement.
Result
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Total Number of Items

Formula first

Overview

The combinations formula, denoted as nCr or (n choose r), determines how many distinct subsets of a given size r can be formed from a larger set of n distinct items. Unlike permutations, the order in which the items are chosen does not matter. This concept is fundamental in probability theory, statistics, and various fields requiring selection without regard to sequence, such as committee formation or lottery odds.

Symbols

Variables

n = Total Number of Items, r = Number of Items to Choose, {}^nC_r = Number of Combinations

Total Number of Items
Number of Items to Choose
Number of Combinations

Apply it well

When To Use

When to use: Use this formula when you need to count the number of ways to select a group of items from a larger collection, and the order of selection is not important. This is common in probability problems, experimental design, and situations where unique groupings are required.

Why it matters: Combinations are crucial for calculating probabilities in scenarios like card games, lotteries, and quality control, where the specific arrangement of chosen items is irrelevant. It helps in understanding the likelihood of events, designing experiments, and making informed decisions in fields ranging from genetics to computer science.

Avoid these traps

Common Mistakes

  • Confusing combinations with permutations (when order matters).
  • Incorrectly calculating factorials, especially for larger numbers.
  • Applying the formula when r > n (which is undefined).

One free problem

Practice Problem

A committee of 3 people is to be chosen from a group of 8 candidates. How many different committees can be formed?

Total Number of Items8 items
Number of Items to Choose3 items

Solve for:

Hint: Use the formula nCr = n! / (r!(n-r)!).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Combination
  2. Discrete Mathematics and Its Applications (Kenneth H. Rosen)
  3. Discrete Mathematics and Its Applications, Kenneth H. Rosen
  4. Wikipedia: Combination (mathematics)
  5. Discrete Mathematics and Its Applications by Kenneth H. Rosen
  6. A First Course in Probability by Sheldon M. Ross
  7. AQA A-Level Mathematics — Statistics (7357/7356)