Multinomial Coefficient Calculator

How it is calculated

Enter the group sizes k₁, k₂, … as non-negative whole numbers separated by commas, spaces or semicolons. The tool adds them to get n = k₁ + k₂ + … (shown as the total) and returns n! ÷ (k₁! × k₂! × …). It builds the answer as a product of ordinary binomial coefficients, C(k₁,k₁) × C(k₁+k₂,k₂) × …, in whole-number arithmetic, so the digits are exact.

Results up to about 9.0 × 10¹⁵ are printed as a full integer. Anything larger switches to scientific notation with four decimals, for example 4.7054e+21, which is an approximation. Decimals are rounded to the nearest whole number (2.5 becomes 3), while negative or non-numeric entries, an empty list, or a total above 1,000 give no result. Groups of size 0 are allowed and change nothing.

Common mistakes

• Entering the item count n as one of the groups. The tool adds the groups itself, so for MISSISSIPPI you enter 1,4,4,2, not 11,1,4,4,2 (which gives 24,443,218,800 for a total of 22).
• Leaving out groups of size one. A single letter such as the M counts as a group of 1; leaving it out makes n one smaller and gives 3,150 instead of 34,650.
• Reading a result in scientific notation as exact. Above about 9 × 10¹⁵ the answer, like 4.7054e+21 for groups 10,10,10,10, is rounded to five significant digits.
• Using the tool when the items are all different. If every item is unique the count is simply n!, and this formula would divide it down by repeats that do not exist.
• Typing decimals such as 2.5. They are rounded to whole numbers, so the group size you meant changes.