\frac10!4! \cdot 5! \cdot 1! - Treasure Valley Movers

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.

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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.

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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:

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