Math & Engineering
Multinomial Coefficient Formula: Group Partitions & Repeated-Item Permutations (FACT)
Calculate the exact multinomial coefficient (n1 + n2 + ... + nk)! / (n1! * n2! * ... * nk!) using FACT and SUM to count ordered squad assignments, categorical outcomes, and word anagrams.
Generalizing combinations from two buckets to k labeled buckets
Binomial combinations n! / (k! * (n - k)!) partition n items into 2 labeled groups (chosen vs unchosen). When partitioning N = n_1 + n_2 + ... + n_k items into k labeled squads of sizes n_1, n_2, ..., n_k — or counting distinct permutations of a string containing repeated letters — the exact count is the multinomial coefficient N! / (n_1! * n_2! * ... * n_k!).
Because FACT(0) = 1, unused zero-sized slots contribute a factor of 1 in the denominator, letting a fixed four-bucket formula FACT(SUM(B2:B5)) / (FACT(B2) * FACT(B3) * FACT(B4) * FACT(B5)) handle 2-, 3-, or 4-category problems seamlessly.
| Problem Scenario | Bucket Counts (n1..n4) | Total N | Denominator Product | Distinct Arrangements |
|---|---|---|---|---|
| Assign 12 engineers to 3 squads (5, 4, 3) | 5, 4, 3, 0 | 12 | 120 * 24 * 6 * 1 = 17,280 | 27,720 |
| Distinct anagrams of "MISSISSIPPI" | 1, 4, 4, 2 | 11 | 1 * 24 * 24 * 2 = 1,152 | 34,650 |
| Roll 6 dice: three 6s, two 5s, one 1 | 3, 2, 1, 0 | 6 | 6 * 2 * 1 * 1 = 12 | 60 |
Exact Multinomial Coefficient With Non-Negative Integer Guard
Computes (B2 + B3 + B4 + B5)! / (B2! * B3! * B4! * B5!) while rejecting negative or fractional counts.
=IF(OR(MIN(B2:B5) < 0, SUMPRODUCT(ABS(B2:B5 - INT(B2:B5))) > 0), "Counts Must Be Non-Negative Integers", FACT(SUM(B2:B5)) / (FACT(B2) * FACT(B3) * FACT(B4) * FACT(B5)))Multinomial Outcome Probability Given Bucket Probabilities p1..p4
Multiplies the multinomial coefficient by p_1^n_1 * p_2^n_2 * p_3^n_3 * p_4^n_4 to compute exact multinomial trial probability.
=ROUND((FACT(SUM(B2:B5)) / (FACT(B2) * FACT(B3) * FACT(B4) * FACT(B5))) * IF(B2 = 0, 1, C2 ^ B2) * IF(B3 = 0, 1, C3 ^ B3) * IF(B4 = 0, 1, C4 ^ B4) * IF(B5 = 0, 1, C5 ^ B5), 6)Try the example
Use the original values or change them to test your understanding.