9! \times \frac5!3! = 362880 \times 20 = 7257600 - Decision Point
Understanding 9! × (5! / 3!) = 7257600: A Step-by-Step Breakdown
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.
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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!
\[
9! = 362880
\]
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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.
