MULTINOMIAL Function (Gnumeric)

If six positions contain three red, two blue and one green item, use =MULTINOMIAL(3,2,1) to count the distinct sequences of those six items. The mathematical expected result is 60, because 6!/(3!×2!×1!) = 720/12. This is independently calculated; this review did not run MULTINOMIAL in Gnumeric.

=MULTINOMIAL(x1,x2,...)

The official Gnumeric Mathematics reference describes x1, x2, and further numeric arguments: it computes the factorial of their sum divided by the product of their individual factorials. In a counting interpretation, each xi represents a non-negative integer number of indistinguishable items of one category. The examples use three or fewer explicitly supplied arguments and do not rely on undocumented optional defaults or array/spill handling. The brief Gnumeric manual does not establish precise Gnumeric runtime coercion or every invalid-input error string, so those are intentionally left for native testing.

The ideal mathematical expression, for whole counts, is:

(x1+x2+...+xm)! / (x1! × x2! × ... × xm!)

For example, counts (2,1,1) mean four positions with two copies of one label and one copy each of the other two labels: 4!/(2!1!1!) = 12. All numeric values below are expected from independent integer calculation.

Type category counts into A2:C7 and enter =MULTINOMIAL(A2,B2,C2) in D2. Fill the formula down. An item count of zero is included as a category with no occurrences.

In row 4, there are 6!/(2!2!2!)=90 unique arrangements. In row 6, AAB, ABA and BAA are the three possible three-position strings when two A items are indistinguishable. These are not probabilities: they are numbers of distinct orderings of a fixed multiset.

For positive small integer counts in A2:C2, a mathematical cross-check is:

=FACT(SUM(A2:C2))/(FACT(A2)*FACT(B2)*FACT(C2))

The right-hand expression gives 60 for A2=3, B2=2 and C2=1 by independent calculation, matching MULTINOMIAL(3,2,1). Because factorials increase quickly, this separate factorial-based check may lose numerical precision for larger inputs; it is an illustrative mathematical identity, not a universal exact-arithmetic guarantee. Gnumeric supports FACT and SUM in its catalogue, but this specific cross-check has not been native-executed for this guide.

A common modelling trap: MULTINOMIAL(3,2,1) counts different orders of six preselected category counts. By contrast, COMBINA(3,6) asks how many unordered multisets of size six could be selected from three available categories (mathematical result 28) and COMBIN(6,3) counts unordered three-item subsets of six distinct items (20). These solve different questions.

• Not a list of groups: This function produces one numeric coefficient, not the individual strings.
• Negative or non-integer counts: The factorial model needs non-negative integer quantities. Gnumeric’s brief published function entry does not spell out the exact native error handling or coercion for negative, fraction or text inputs, so do not transplant Excel-specific #NUM!/#VALUE! messages without application testing.
• Too many inputs or cell ranges: MULTINOMIAL(x1,x2,...) is a variadic notation. This guide only demonstrates separate cell arguments; it does not claim range arguments, empty-cell coercion, maximum argument count or array evaluation are supported in every build.
• Large results: A coefficient may outgrow the display precision even when mathematically an integer. Avoid using rounded spreadsheet values as exact combinatorial identifiers without additional validation.
• Argument order: Categories can be supplied in any order mathematically; the multiset coefficient is symmetric. This is an independent mathematical property, not a claim about errors when inputs are malformed.
• Locale: The formula examples use English comma separators; verify formula parsing on your installed Gnumeric locale.

Gnumeric’s official documentation marks MULTINOMIAL as Excel compatible. Microsoft also publishes MULTINOMIAL (its current example is localized; the Gnumeric formulas above deliberately use English separators). Both products document the factorial-ratio concept, but this does not establish identical coercion, argument limits or numerical precision. Related Gnumeric pages: COMBIN, COMBINA, FACT, SUM and PERMUT.

COMBIN, COMBINA, FACT, SUM, PERMUT.

The official Gnumeric reference establishes the function’s purpose and argument structure. The arguments and worked examples were editorially reviewed and the stated expected results were independently calculated.

Review date: 2026-10-12. No native application run: the worked outputs are documentation-based or independently calculated expectations, not measurements from installed Gnumeric. Ordinary numeric cases and the stated text-construction examples are checked; blank/text coercion, machine extremes and workbook round trips are not certified. The formula editor in your installed version is the authority for localized separators.

Example-check record · Editorial standards.