Multinomial Coefficient Calculator Without Repeats

A multinomial coefficient without repeats calculates the number of ways to partition a set of distinct items into distinct groups of specified sizes. This calculator helps you compute these coefficients efficiently.

What is a multinomial coefficient?

A multinomial coefficient is a generalization of the binomial coefficient. While binomial coefficients count the number of ways to choose 2 items from a set, multinomial coefficients count the number of ways to partition a set into multiple groups of specified sizes.

For example, if you have 5 distinct items and want to divide them into groups of 2, 2, and 1, the multinomial coefficient would tell you how many distinct ways this can be done.

Key difference: Multinomial coefficients without repeats assume all items are distinct and each group size is unique.

Formula and calculation

The multinomial coefficient for partitioning n distinct items into groups of sizes k₁, k₂, ..., kₘ is given by:

M(n; k₁, k₂, ..., kₘ) = n! / (k₁! × k₂! × ... × kₘ!)

Where:

n is the total number of distinct items
k₁, k₂, ..., kₘ are the sizes of the groups
! denotes factorial

This formula assumes that the sum of all group sizes equals n (k₁ + k₂ + ... + kₘ = n) and that all group sizes are positive integers.

Worked examples

Example 1: Simple partition

Suppose you have 4 distinct books and want to divide them into groups of 2 and 2.

M(4; 2, 2) = 4! / (2! × 2!) = 24 / (2 × 2) = 6

There are 6 distinct ways to partition 4 books into two groups of 2 each.

Example 2: Three-group partition

With 6 distinct students, you want to divide them into groups of 2, 3, and 1.

M(6; 2, 3, 1) = 6! / (2! × 3! × 1!) = 720 / (2 × 6 × 1) = 60

There are 60 distinct ways to partition 6 students into groups of 2, 3, and 1.

Comparison of multinomial coefficients
Total items (n)	Group sizes	Multinomial coefficient
5	2, 2, 1	30
6	3, 2, 1	60
7	3, 2, 2	210

Applications

Multinomial coefficients without repeats are used in various fields:

Combinatorics: Counting distinct arrangements and partitions
Probability: Calculating probabilities of specific outcomes in multinomial distributions
Statistics: Analyzing experimental designs with multiple factors
Computer Science: Algorithms for partitioning problems

Understanding multinomial coefficients helps in solving problems where items need to be divided into distinct groups of specific sizes.

FAQ

How is this different from a binomial coefficient?

A binomial coefficient counts ways to choose 2 items from a set, while a multinomial coefficient counts ways to partition into multiple groups of specified sizes.

Can the group sizes be the same?

Yes, but the items must be distinct. For example, partitioning 4 items into two groups of 2 each is valid.

What if the sum of group sizes doesn't equal n?

The multinomial coefficient is undefined in this case. The sum of all group sizes must exactly equal the total number of items.