\frac10!3! \cdot 5! \cdot 2! - Treasure Valley Movers

Understanding the Factorial Expression: \(\frac{10!}{3! \cdot 5! \cdot 2!}\)

When exploring combinatorics and probability, factorials often play a central role. One intriguing expression is:

\[
\frac{10!}{3! \cdot 5! \cdot 2!}
\]

At first glance, this fraction may appear abstract, but it encodes meaningful mathematical and practical significance—especially in counting problems. Let’s break down what this expression means, simplify it, and explore its significance.

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What Does the Factorial Expression Mean?

Factorials represent the product of all positive integers up to a given number. For example:

• \(10! = 10 \ imes 9 \ imes 8 \ imes \cdots \ imes 1\)
  - \(3! = 6\), \(5! = 120\), \(2! = 2\)

So, the given ratio:

\[
\frac{10!}{3! \cdot 5! \cdot 2!}
\]

can be interpreted as the number of distinct ways to partition a set of 10 objects into three labeled groups of sizes 3, 5, and 2, respectively. This is a multinomial coefficient, often denoted:

\[
\binom{10}{3, 5, 2} = \frac{10!}{3! \cdot 5! \cdot 2!}
\]

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How to Simplify and Compute the Expression

Let’s simplify step-by-step:

\[
\frac{10!}{3! \cdot 5! \cdot 2!} = \frac{10 \ imes 9 \ imes 8 \ imes 7 \ imes 6 \ imes 5!}{3! \cdot 5! \cdot 2!} = \frac{10 \ imes 9 \ imes 8 \ imes 7 \ imes 6}{3! \cdot 2!}
\]

Now compute the numerator:

\[
10 \ imes 9 = 90,\quad 90 \ imes 8 = 720,\quad 720 \ imes 7 = 5040,\quad 5040 \ imes 6 = 30240
\]

Numerator = \(30240\)

Denominator:

\[
3! = 6,\quad 2! = 2 \quad \Rightarrow \quad 6 \cdot 2 = 12
\]

Now divide: