COMBINA in Google Sheets: Repeated Choices and Stars-and-Bars Counts

• Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
• Availability: documented in the rolling Google Sheets service. Examples use ordinary web-spreadsheet numeric cells; other clients and imported files are not separately certified.
• Checked: October 14, 2026; official COMBINA reference. Documentation and independent calculations, without native Sheets execution.

=COMBINA(n,k)

Google requires nonnegative n and k, describes order as irrelevant, and explicitly truncates decimal parts before calculation. The worked examples use positive type counts and small whole selection counts so the underlying counting model is clear.

The help states that an argument sum n+k at least 1031 produces #NUM!. That is a vendor-documented restriction, not an independently executed boundary test. The all-zero and zero-type configurations are not resolved by the positive-type factorial identity used here.

For two types A and B and three selected units, the four mixtures are AAA, AAB, ABB and BBB:

=COMBINA(2,3)

Expected count: 4. ABA is not added separately because it has the same type counts as AAB. If all three positions were ordered instead, there would be 2^3=8 sequences.

This Sheets example follows Google’s unrestricted nonnegative n and k description. It does not certify a different product’s documentation or disputed argument restrictions merely because the function name is shared.

With three types, enter selected counts 1, 2, 3 and 4 in A2:A5. Fill =COMBINA(3,A2) down.

Expected outputs are 3, 6, 10 and 15. For a positive truncation control, =COMBINA(3.9,2.8) is expected to use n=3 and k=2 and return 6. Fractional counts may still be unacceptable in the practical source data, even though truncation is documented.

Google’s prose prints the abbreviated expression COMBIN(n+k−1), while its accompanying choose-k description and factorial formula identify the second argument. The complete worksheet identity for positive n is:

=COMBIN(n+k-1,k)

Here n and k are explanatory placeholders. For the concrete two-type/three-unit example, =COMBIN(2+3-1,3) gives 4, matching COMBINA(2,3). Copying the incomplete one-argument text would omit essential syntax.

Place four identical tokens into three labeled boxes, allowing empty boxes and imposing no capacity limit. Each outcome is determined by three nonnegative box counts that sum to four.

=COMBINA(3,4)

Expected distribution count: 15. An independent enumeration groups by the first box’s count: 0, 1, 2, 3 or 4 leaves 5, 4, 3, 2 or 1 allocations for the other two boxes, totaling 15.

The tokens are indistinguishable, while boxes are labeled. Requiring a token in every box or putting a capacity limit on a box changes the allowed outcomes and requires an adjusted model.

Among three-unit mixtures from three types A, B and C, there are 10 total mixtures. The mixtures containing no A use only B and C and number 4.

=COMBINA(3,3)-COMBINA(2,3)

Expected count containing at least one A: 6. Subtraction is valid because the excluded mixtures are a subset of the same total outcome space.

The factorial construction for positive n is FACT(n+k−1)/(FACT(k)×FACT(n−1)). Google notes that COMBINA supports larger inputs than that expanded construction; do not assume the expansion is a safe replacement near large magnitudes.

The examples and enumeration verify finite mathematical expectations. No native zero-type, full error-coercion or cross-client execution matrix is claimed.

• Google COMBINA help.
• Editorial testing standards.

Inputs are fictional. Approximate numerical answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Complete coercion, extreme-magnitude and cross-client behavior remain untested.