Understanding 9! × (5! / 3!) = 7257600: A Step-by-Step Breakdown

Mathematics often reveals elegant simplifications behind seemingly complex expressions. One such intriguing equation is:
9! × (5! / 3!) = 7257600

At first glance, factorials and division might seem intimidating, but once broken down, this equation showcases beautiful algebraic structure and the power of breaking down large computations. In this article, we’ll explore how this equality holds true, step by step.

Understanding the Context

What Are Factorials?

A factorial, denoted by n!, represents the product of all positive integers from 1 to n:
- \( 3! = 3 × 2 × 1 = 6 \)
- \( 5! = 5 × 4 × 3 × 2 × 1 = 120 \)
- \( 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362880 \)

Factorials grow extremely fast, so understanding how they interact in equations is key to solving factorial expressions correctly.

=3 \times \frac{22}{7} \times(3.5)^{2}=\frac{66}{7} \times 12.25 \\ | Filo

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=3 \times \frac{22}{7} \times(3.5)^{2}=\frac{66}{7} \times 12.25 \\ | Filo
Solved 9.2 (+ - 0.7) times ([5.4 (+ - 0.3) times 10-3] + | Chegg.com
Simply i. \(\frac{6^8\times5^3}{10^3\times3^4}\) - Sarthaks eConnect ...
Simply i. \(\frac{6^8\times5^3}{10^3\times3^4}\) - Sarthaks eConnect ...
Solved 1. Multiply the following: (i) 65×30 (ii) 98×81 (iii) | Chegg.com

Key Insights

Breaking Down the Equation: 9! × (5! / 3!)

We evaluate the expression:
\[
9! \ imes \frac{5!}{3!}
\]

Let’s isolate each component:

Step 1: Calculate 9!
\[
9! = 362880
\]

Continue Reading

Substitute \( a \) and \( b \) into \( 7a + 3b + c = -4 \):
7\left(-\frac{7}{6}\right) + 3\left(\frac{21}{2}\right) + c = -4
-\frac{49}{6} + \frac{63}{2} + c = -4

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Final Thoughts

Step 2: Compute 5! and 3!
\[
5! = 120 \
3! = 6
\]

Step 3: Evaluate the quotient \( \frac{5!}{3!} \)
\[
\frac{5!}{3!} = \frac{120}{6} = 20
\]

Putting It All Together

Now substitute the computed values:
\[
9! \ imes \frac{5!}{3!} = 362880 \ imes 20 = 7257600
\]

This confirms the original equation:
9! × (5! / 3!) = 7257600

Why This Factorial Manipulation Matters

While only a computation, this expression highlights:

  • Modular arithmetic simplification: Dividing smaller factorials first reduces computational complexity.
    - Pattern recognition: Understanding how factorial division works enables faster mental math.
    - Applications in combinatorics: Expressions like this often appear in permutations and probability problems.