Combination Calculator

A combination calculator finds n choose r, written nCr: the number of ways to choose r distinct items from n distinct items when selection order does not matter and items are not repeated. Choosing people for a committee is a standard example because the same members form the same committee regardless of listing order.

The implementation truncates inputs to integers and requires 0 ≤ r ≤ n ≤ 1000. It uses a multiplicative formula rather than directly calculating large factorials, then rounds the numerical result to the nearest integer.

Combination counts grow rapidly, making manual enumeration impractical. The calculator supports counting problems and provides inputs for probability models when every eligible subset is equally likely.

The numeric answer alone does not establish equal likelihood, independence, or any probability assumption. It only counts unordered selections under the stated distinct-item, no-repetition model.

• Count unordered selections.
• Avoid direct full-factorial calculation.
• Use symmetry between r and n − r.

Enter the total number of available items as n and the number chosen as r. The algorithm uses k = min(r, n − r), then multiplies and divides through k terms.

Using the smaller side reduces work because choosing r included items uniquely determines the n − r excluded items. Inputs outside the allowed range return no result.

The output is a count of distinct subsets. For n = 10 and r = 3, the result 120 means there are 120 different three-item groups, not 120 ordered sequences.

nC0 and nCn both equal 1. The result assumes all n items are distinguishable. If items are identical, repetition is allowed, or constraints apply, another counting model is needed.

Distinguishing combinations from permutations prevents overcounting by a factor related to r!. Many probability errors begin by counting arrangements when only membership matters.

Large results can exceed JavaScript's exact-integer precision. The tool allows n through 1000, but a displayed finite number at large values may be rounded or eventually represented in exponential notation or become nonfinite.

The mathematical factorial formula is n!/[r!(n−r)!]. The implementation evaluates an equivalent product over the smaller of r and n−r, reducing unnecessary intermediate work.

nCr = n! ÷ (r! × (n − r)!), for integers 0 ≤ r ≤ n

n

The total number of distinct available items, truncated to an integer.

r

The number of distinct items selected, truncated to an integer.

!

Factorial: the product of positive integers down to 1.

• Order does not matter.
• Selections are modeled without replacement.
• Items are assumed distinct for counting purposes.

The combination calculator counts unordered, no-repetition selections of distinct items with 0 ≤ r ≤ n ≤ 1000. Confirm that order truly does not matter and use a big-integer system when exactness for extremely large counts is essential.