Combinations (nCr) Calculator

How it is calculated

The number of ways to pick r items from n distinct items, ignoring order, is C(n, r) = n! ÷ (r! × (n − r)!). The tool avoids the huge factorials: it multiplies step by step, result = result × (n − i) ÷ (i + 1) for i from 0 to k − 1, where k is the smaller of r and n − r, because C(n, r) = C(n, n − r). While the value fits in a 64-bit integer (up to about 9.2 × 10¹⁸) the answer is an exact whole number with thousands separators. Beyond that it is shown in scientific notation with 5 significant digits, for example 1.0089 × 10²⁹ for C(100, 50).

Both n and r must be whole numbers with 0 ≤ r ≤ n, and n can be at most 170. A decimal, a negative number, r larger than n or n above 170 shows no result.

Common mistakes

• Using nCr when order matters. Choosing a president, a secretary and a treasurer from 10 people is a permutation (720 ways), not C(10, 3) = 120.
• Swapping n and r. n is the size of the whole set and r the number you pick; with r larger than n the tool shows nothing, since you cannot pick more items than exist.
• Entering a probability or percentage instead of a count. n and r must be whole numbers of items.
• Forgetting that repeats are not allowed. nCr assumes each item can be picked once, which is not the case for drawing with replacement.
• Reading a result like 1.0089 × 10²⁹ as a rounded-off whole number. It is a 30-digit number of which only the first five digits are shown.