Combination Calculator

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C(n,r)

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Table of Contents

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What is Combination Calculator?

A combination counts how many groups of a chosen size can be formed from a larger set when the order within each group does not matter. Choosing three people for a committee is a combination problem if those people have identical roles. Listing the same three names in a different order does not create another committee.

This calculator uses the standard formula for selections without replacement from distinct items. It is useful for small counting exercises, selection planning, and checking the denominator in a simple probability calculation. Before entering numbers, decide whether your problem actually treats order and repeated selections that way.

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How to Use This Calculator

Enter the total number of available items in n and the number selected in r. Both must be nonnegative whole numbers, and r cannot exceed n. For choosing three people from five, enter n = 5 and r = 3, then select Calculate.

The current form does not fully enforce those mathematical restrictions, so check them yourself. A decimal selection size, a negative value, or r greater than n is outside this formula’s intended domain even if the page returns a number. The result is displayed as a whole number with comma separators.

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Formula Used

C(n,r) = n! ÷ [r! × (n − r)!]. The factorial n! is the product of the positive integers from 1 through n, with 0! defined as 1. The notation “n choose r” refers to this same quantity.

One way to see the formula is to first count ordered selections: n × (n − 1) × … × (n − r + 1). Each unordered group appears r! times in that ordered count, once for each arrangement of its selected members. Dividing by r! removes those repetitions.

The implementation computes factorials directly. Factorials grow quickly, so large inputs can exceed ordinary floating-point capacity or lose integer precision. A long answer formatted with commas is not evidence that every digit remains exact.

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Variables Explained

Symbol or inputMeaning
nTotal number of distinct items available. Use a nonnegative integer.
rNumber of items chosen for each group. Use a whole number from 0 through n.
n!Factorial of n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
C(n,r)Number of different unordered selections without replacement. This is a count, not a percentage.
n − rNumber left unselected. Choosing the omitted items identifies the same group in another way.
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Worked Example

For five people A, B, C, D, and E, choosing three gives C(5,3) = 5! ÷ (3! × 2!) = 120 ÷ (6 × 2) = 10. The groups are ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, and CDE.

ABC and CBA are the same group because the selected people are identical. If those three places instead represented president, secretary, and treasurer, their order would assign different jobs. That would be a permutation problem and would have 5 × 4 × 3 = 60 outcomes.

As a second check, C(8,2) = 8 × 7 ÷ 2 = 28. You can calculate this smaller product by hand without writing out all of 8!, which is often more convenient for checking a result.

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Why Use This Calculator?

The output counts possible groups under the stated rules. It does not tell you which group to choose or whether each group is equally likely in a real selection process. If you use it in probability, identify favorable outcomes and all possible outcomes separately, then justify why those outcomes have equal chances.

The symmetry C(n,r) = C(n,n − r) provides a quick check. Choosing three of five is equivalent to identifying the two left out, so both counts are ten. The boundary cases also make sense: there is exactly one way to choose nobody and one way to choose the entire set.

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Common Mistakes

    Do not use combinations for ranked finishers, passwords, seating orders, or assigned roles unless the ordering truly does not matter. Do not apply the same formula to drawing an item, replacing it, and drawing again; replacement changes the allowed selections.

    Also distinguish distinct physical items from repeated categories. Choosing two specific red balls from a collection of separately identifiable balls is different from counting color patterns when all red balls are treated as interchangeable. Define the objects before inserting their count into n. For large exact counts, use a method designed for arbitrary-precision integers rather than trusting the factorial implementation here.

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Frequently Asked Questions

Why is choosing zero items counted as one way?

There is one possible empty selection. The formula agrees because C(n,0) = n! ÷ (1 × n!) = 1. It counts possible selections, not the number of objects inside a selection.

Can the answer be a fraction?

For valid integer inputs in this model, the exact answer is always a nonnegative integer. A fractional intermediate result or an invalid-domain answer should prompt a check of the inputs and arithmetic. Whole-number formatting alone does not establish validity.

Does the calculator allow repeated selections?

The formula is for selection without replacement. An item cannot occupy two places in the same group. Problems that explicitly allow repetition require a different counting model and should not be entered as ordinary combinations.

Why do n choose r and n choose n minus r match?

Every selected group determines one complementary group of omitted items, and vice versa. This one-to-one relationship means the two counts are equal. For example, choosing two out of eight and leaving out six both describe 28 possibilities.

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Sources & References

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