Count arrangements of a multiset from the multiplicities of its repeated groups. Each submitted value produces distinct arrangements plus the intermediate reasoning.
An independent Multinomial Coefficient pass needs Group counts plus the fixed condition. Judge whether Distinct arrangements has a plausible sign and scale. Test the fixed condition separately; otherwise the cause of a changed Multinomial Coefficient Distinct arrangements remains unclear.
For group counts n₁,…,nᵣ totaling n, the multinomial coefficient n!/(n₁!…nᵣ!) counts strings when objects within each group are indistinguishable. Multinomial Coefficient can also be compared with repeatable position choices.
Every count must be a nonnegative integer. The model assumes group membership is the only source of indistinguishability and that the counts account for all positions.
It counts repeated-letter arrangements, category assignments, grouped outcomes, and coefficients in multinomial expansions. A related application of Multinomial Coefficient is two-group case.
Counts 4,3,2 total 9, so the number of distinct arrangements is 9!/(4!3!2!)=1,260.
Add the group sizes, take the factorial of the total, then divide by a factorial for every group to remove rearrangements that look identical.
Test Multinomial Coefficient first with modest values for Group counts that can be evaluated without software. A tree, short table, Pascal row, or handwritten outcome list supplies a structurally different comparison and makes a misplaced factorial or complement visible.
For Multinomial Coefficient, next inspect the allowable endpoints. Probabilities must stay between zero and one, counts cannot become negative, and a requested subset cannot exceed its parent. Passing those checks supports the setup; it does not by itself prove that the real situation satisfies the model assumptions.
Substitute the Multinomial Coefficient solution into Group counts. This Multinomial Coefficient check rejects false Multinomial Coefficient branches and forbidden denominators.
Choose an easy Group counts value before running Multinomial Coefficient. Predict the result, then compare it with the Multinomial Coefficient output.
Their sum defines n.
Yes; its 0! factor is 1.
Swaps inside an identical group do not create new arrangements.