Mathematics calculator

Permutation and Combination Calculator

Count ordered arrangements and unordered selections of r objects chosen from a set of n distinct objects.

Calculator inputs

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Counting methods
Calculated output

Result summary

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Complete the required fields and select the calculate button.

Table of Contents

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What is Permutation and Combination?

The Permutation and Combination Calculator compares two ways of counting selections from distinct objects. A permutation counts ordered selections, while a combination counts groups for which order is irrelevant. Seeing both values together helps expose the effect of treating different orders as separate outcomes.

This page is intended for large counting problems. Instead of building enormous factorials directly, it adds their base-10 logarithms and displays each answer in scientific notation with two decimal places. That keeps the output readable, although the shown mantissa is rounded and may not reproduce every digit of the exact integer.

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How to Use This Calculator

Enter the total number of objects as n and the sample size as r. Use whole numbers from 0 through 10,000, with r no larger than n. Select Calculate to see the combination and permutation counts. Both formulas assume distinct objects, selection without replacement and no extra restrictions.

Choose which output fits the question by trying a swap. If selecting A then B is different from selecting B then A, use the permutation. If both descriptions refer to the same selected group, use the combination. A podium finish is ordered; a two-person committee with equal roles is not.

The displayed form “a × 10^b” means a multiplied by ten raised to exponent b. For small examples, 2.00 × 10^1 means 20. Because the display keeps two digits after the decimal point, use another exact-integer method when every digit of a large count is required.

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Formula Used

Permutation: P(n,r) = n! ÷ (n − r)!

Combination: C(n,r) = n! ÷ [r!(n − r)!]

Equivalently, P(n,r) = C(n,r) × r!. The page evaluates the logarithms of the factorial terms, then converts each resulting logarithm to scientific notation.

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Variables Explained

Symbol or displayMeaningPage rule
nNumber of distinct available objectsWhole number from 0 to 10,000
rNumber selected from those objectsWhole number from 0 through n
P(n,r)Selections where order mattersNo object is reused
C(n,r)Selections where order does not matterNo object is reused
a × 10^bScientific notationMantissa a is displayed to two decimals

The word sample here means the number chosen, not a statistical sample with probabilities or confidence limits. The calculator supplies counts only.

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Worked Example

Suppose five distinct people are available and two are selected. For assigned roles such as chair and secretary, order matters:

P(5,2) = 5! ÷ 3! = 5 × 4 = 20

For a two-person committee with equal roles, order does not matter:

C(5,2) = 5! ÷ (2!3!) = 10

The page displays these as approximately 2.00 × 10^1 permutations and 1.00 × 10^1 combinations. Their ratio is 2! = 2 because every two-person group can be arranged in two orders.

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Why Use This Calculator?

Use the combination result when the chosen members alone define an outcome. Use the permutation result when their positions, sequence or roles are part of the outcome. The formulas can both be mathematically correct while only one matches the real question.

The scientific display communicates scale efficiently. For example, an exponent of 12 means the result is on the order of a trillion. Since the mantissa is rounded to two decimal places, the displayed figure is not an exact record of a very large count. Small examples can be checked by factorial cancellation or by listing possibilities.

These counts often appear in probability denominators, but they do not establish a probability on their own. Confirm that outcomes are equally likely and count the favorable outcomes using compatible assumptions before forming a ratio.

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Common Mistakes

  • Choosing permutation merely because objects were drawn one after another, even though only the final group matters.
  • Choosing combination when selected positions have different titles or ranks.
  • Using r greater than n in a selection without replacement.
  • Entering fractions for counts of objects.
  • Reading a rounded scientific result as an exact value with every digit known.
  • Applying the formulas to identical objects or conditional selection rules without adjustment.
  • Using the count itself as a probability.
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Frequently Asked Questions

How do I decide between permutation and combination?

Ask whether changing the order of the same selected objects creates a new outcome. If it does, read the permutation. If the selected group remains the same, read the combination.

Why does the page show powers of ten for small answers?

This calculator uses one scientific display for every valid size. Thus 20 appears as 2.00 × 10^1. Move the decimal one place to the right to read that particular result as 20.

Are the displayed counts exact?

The mathematical counts are integers, but this page derives their logarithms and rounds the scientific mantissa to two decimal places. The display preserves useful scale, not all digits of a large exact count.

Why is the permutation count related by r factorial?

Each combination selects one unordered group of r distinct objects. That group can be arranged in r! orders, so multiplying the combination count by r! gives the corresponding permutation count.

Does the calculator allow an object to be chosen twice?

No. Both displayed formulas describe selection without replacement. A sequence where every choice can recur uses n raised to r instead of the permutation formula shown here.

Can I use these counts directly for probability?

Only after confirming that the counted outcomes form the appropriate equally likely sample space. You must also count favorable outcomes under the same order and replacement rules, then divide favorable by total.

Why are zero objects or zero selections accepted?

Standard counting conventions define 0! as 1. There is one empty selection, so both P(n,0) and C(n,0) equal one; when n and r are both zero, the same convention applies.

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Sources & References