combinations without repetition calculator

What is a combination without repetition?

A combination without repetition counts how many unique groups you can form from a larger set, where:

• Order does not matter (A,B,C is the same as C,B,A)
• No item can repeat (once an item is chosen, it cannot be chosen again)

This is also called combinations without replacement, n choose r, or the binomial coefficient.

Formula used by this calculator

The calculator uses the standard combination formula:

C(n, r) = n! / (r! × (n - r)!)

Where:

• n = total number of available items
• r = number of items selected
• ! = factorial (for example, 5! = 5×4×3×2×1)

For performance and accuracy, this page computes nCr directly using integer math (BigInt), which avoids floating-point rounding issues for large values.

How to use the combinations calculator

1. Enter the total number of items in the first field (n).
2. Enter how many items you want to choose in the second field (r).
3. Click Calculate C(n, r).
4. Read the total number of unique combinations in the result box.

If your result is very large, the calculator also shows the number of digits and a scientific-notation style preview.

Examples of combinations without repetition

Example 1: Poker hands

How many 5-card hands can be dealt from a 52-card deck?

C(52,5) = 2,598,960

That means there are 2,598,960 unique 5-card combinations.

Example 2: Choosing a committee

If you choose 4 people from 12 candidates:

C(12,4) = 495

So there are 495 possible committees.

Example 3: Lottery-style selection

Choose 6 numbers from 49 total numbers:

C(49,6) = 13,983,816

This is a common way to compute lottery odds.

Combinations vs permutations

A common confusion in probability is when to use combinations or permutations:

• Combinations: use when order does not matter.
• Permutations: use when order does matter.

For example, selecting 3 winners from 10 contestants is a combinations problem. Assigning gold, silver, and bronze to 3 people is a permutations problem because rank order matters.

Common mistakes to avoid

• Entering decimals (n and r should be whole numbers only).
• Using r > n (not possible in combination without repetition).
• Using combinations when your problem actually depends on order.
• Assuming repetition is allowed (this tool is specifically without repetition).

Quick FAQ

Can r be 0?

Yes. C(n,0) = 1 because there is exactly one way to choose nothing.

Can r equal n?

Yes. C(n,n) = 1 because there is exactly one way to choose all items.

Why are results so large?

Binomial coefficients grow very quickly. Even moderate values of n and r can produce massive integers, especially around r ≈ n/2.

Final note

This combinations without repetition calculator is ideal for probability homework, interview prep, data science intuition, lottery math, card game analysis, and any situation where you need an accurate nCr calculator.