\frac{10!}{4! \cdot 5! \cdot 1!}

February 17, 2026

$\frac{10!}{4! \cdot 5! \cdot 1!}$

Understanding the Combinatorial Expression: $\frac{10!}{4! \cdot 5! \cdot 1!}$

When exploring advanced mathematics and combinatorics, expressions involving factorials often appear in probability, statistics, and counting problems. One such fascinating expression is:

$$\frac{10!}{4! \cdot 5! \cdot 1!}$$

At first glance, this fraction might seem abstract, but it represents a well-defined mathematical quantity with clear real-world interpretations. In this article, we'll break down this combinatorial expression, explain its mathematical meaning, demonstrate its calculation steps, and highlight its significance in combinatorics and practical applications.

What Is This Expression?

This expression is a form of a multinomial coefficient, which generalizes the concept of combinations for partitioning a set into multiple groups with specified sizes. Here:

$$\frac{10!}{4! \cdot 5! \cdot 1!}$$

is equivalent to the number of ways to divide 10 distinct items into three distinct groups of sizes 4, 5, and 1 respectively, where the order within each group does not matter, but the group labels do.

Although $1!$ may seem redundant (since $x! = 1$ for $x = 1$), explicitly including it maintains clarity in formal combinatorial notation.

Step-by-Step Calculation

To compute this value, let's evaluate it step by step using factorial properties:

Step 1: Write out the factorials explicitly

$$10! = 10 \ imes 9 \ imes 8 \ imes 7 \ imes 6 \ imes 5!$$This allows cancellation with $5!$ in the denominator.

So:

$$\frac{10!}{4! \cdot 5! \cdot 1!} = \frac{10 \ imes 9 \ imes 8 \ imes 7 \ imes 6 \ imes 5!}{4! \cdot 5! \cdot 1}$$

Cancel $5!$:

$$= \frac{10 \ imes 9 \ imes 8 \ imes 7 \ imes 6}{4! \cdot 1}$$

Now compute $4! = 4 \ imes 3 \ imes 2 \ imes 1 = 24$

Then:

$$= \frac{10 \ imes 9 \ imes 8 \ imes 7 \ imes 6}{24}$$

Compute numerator step-by-step:

$10 \ imes 9 = 90$- $90 \ imes 8 = 720$- $720 \ imes 7 = 5040$- $5040 \ imes 6 = 30240$

Now divide:

$$\frac{30240}{24} = 1260$$

Final Answer:

$$\frac{10!}{4! \cdot 5! \cdot 1!} = 1260$$

Mathematical Meaning: Multinomial Coefficient

This expression represents the number of distinct ways to partition 10 labeled objects into three labeled groups of sizes 4, 5, and 1.

We can interpret this combinatorially:- Choose 4 out of 10 items for the first group- Then choose 5 out of the remaining 6 for the second group- The final 1 item automatically goes to the third group

Using the multinomial formula:

$$\binom{10}{4, 5, 1} = \frac{10!}{4! \cdot 5! \cdot 1!} = 1260$$

This coefficient appears when counting permutations where certain blocks are grouped and treated distinctly—critical in statistics, probability, and partitioning problems.

Practical Applications

1. Probability and Sampling

In sampling without replacement, such coefficients help compute the number of ways to form specific sample groups. For example, if a committee of 10 people is divided into smaller teams, this formula quantifies team compositions.

2. Statistical Physics and Chemistry

In partitioning particles into energy states or distributing molecules into compartments, multinomial coefficients model distinct groupings.

3. Computer Science and Databases

Used in data partitioning, shuffling, or assigning tasks across processors when load balancing or labeling is required.

Connections to Special Functions

While not a central constant, this coefficient relates to permutations with repetition where group counts define redundancy. It also appears in generating functions for combinatorial sequences and in inclusion-exclusion principles.

Summary

The expression: