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:

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

Understanding the Context

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:

$SOLVED:Sum the series 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+3 \cdot 4 ...$

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$SOLVED:Sum the series 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+3 \cdot 4 ...$

$SOLVED:Simplify each expression. a. \frac{3 \cdot 5 \cdot x \cdot y ...$

$SOLVED:Determine if the series 1-(1 / 2)+\{(1 \cdot 3) /(2 \cdot 4 ...$

$SOLVED:Sum the series 1 \cdot 2 \cdot 5+2 \cdot 3 \cdot 6+3 \cdot 4 ...$

$algebra precalculus - Calculate $\frac {1}{2\cdot3\cdot 4}+\frac {1}{3 ...$

Key Insights

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

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

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Final Thoughts

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:

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