Multinomial Counting
Arrange items with repeated types: divide by the repeats.
What is it?
The number of orderings of items with identical copies of each type is the multinomial coefficient.
Why is it useful?
Word rearrangements, lattice paths and splitting people into labeled groups all use it.
When should you think of it?
Think of it when items repeat, or when a set is split into labeled groups of given sizes.
Think of this method when…
- Rearranging letters of a word with repeats
- Splitting people into groups of fixed sizes
- Lattice paths with fixed numbers of each step
- Probability of specific type counts in trials
Core idea
Each distinct arrangement corresponds to labeled ones.
Step-by-step playbook
- 1Count the total and each type's multiplicity.
- 2Compute .
- 3Divide further if groups are unlabeled and equal-sized.
Worked example
Worked example
How many distinct arrangements are there of the letters of MISSISSIPPI?
- 1
11 letters: I ×4, S ×4, P ×2, M ×1.
- 2
Apply the multinomial.
- Answer:
Common mistakes
- Forgetting to divide when equal-sized groups are unlabeled.
- Treating distinct items as identical.
Would you use this method?
How many ways are there to place 7 identical balls into 3 boxes?
Practice this method
8 QuantSolve questions are tagged with this method.
Practice this method