Combination Calculator

Calculate selections where order doesn't matter: C(n,r)

Total items (n)

Items to select (r)

How to Calculate Combinations

Enter Total Items (n)

Input the total number of items in your set from which you're selecting.

Enter Items to Select (r)

Specify how many items you want to choose from the total set.

Get Combination Result

See the total number of possible combinations with the complete formula breakdown.

Understanding Combinations in Mathematics

📐 Order Doesn't Matter

Combinations count selections where order is irrelevant. Choosing {A,B} is the same as {B,A} in combinations.

🎲 Probability & Statistics

Essential for calculating odds in lottery, card games, and statistical sampling problems.

🔬 Scientific Applications

Used in genetics, chemistry, and research for calculating possible outcomes and sample sizes.

📚 Step-by-Step Formula

See the complete factorial calculation showing how C(n,r) is derived using n! / (r! × (n-r)!).

Combinations vs Permutations

Feature	Combinations	Permutations
Order Matters?	No	Yes
Formula	C(n,r) = n!/(r!(n-r)!)	P(n,r) = n!/(n-r)!
Example	Committee selection	Race rankings
Result Size	Smaller (divides by r!)	Larger

Combination Formula FAQs

What is the combination formula?

C(n,r) = n! / (r! × (n-r)!) where n is total items, r is items to select, and ! means factorial (multiply all integers up to that number).

When do I use combinations instead of permutations?

Use combinations when order doesn't matter (picking team members, lottery numbers). Use permutations when order matters (race results, passwords).

How many combinations of 3 from 10?

C(10,3) = 10!/(3!×7!) = (10×9×8)/(3×2×1) = 720/6 = 120 possible combinations.

What does nCr mean in math?

nCr is another notation for combinations, read as "n choose r". It's the same as C(n,r) and represents choosing r items from n total.

Can r be larger than n in combinations?

No, you cannot select more items than available. If r > n, the combination equals 0 (impossible selection).

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