\frac50406 \cdot 2 \cdot 2 = \frac504024 = 210

Understanding the Math: \frac{5040}{6 \cdot 2 \cdot 2} = \frac{5040}{24} = 210

When tackling fractions and arithmetic formulas, simplifying complex expressions can reveal elegant mathematical truths. One such expression is:

\[
\frac{5040}{6 \cdot 2 \cdot 2} = \frac{5040}{24} = 210
\]

Understanding the Context

This breakdown demonstrates how multiplication in the denominator simplifies a larger number division—ultimately revealing why this equation holds true.

Breaking Down the Expression

At the heart of this calculation is division by a product of numbers. Let’s examine each step carefully:

The denominator is expressed as $6 \cdot 2 \cdot 2$. Starting with multiplication from left to right:

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

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

$If \( \frac{1}{2 \cdot 3^{10}}+\frac{2}{2^{2} \cdot 3^{9}}+\frac{3}{2 ...$

$sequences and series - Prove that $\frac{5}{1 \cdot 2 \cdot 3} + \frac ...$

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Exercise 6.7.6. (a) Let cn=2⋅4⋅6⋯2n1⋅3⋅5⋯(2n−1) for | Chegg.com

Key Insights

First, compute $6 \ imes 2 = 12$
- Then multiply the result by 2 again: $12 \ imes 2 = 24$

So the original fraction becomes:

\[
\frac{5040}{24}
\]

Why 5040?

The numerator, 5040, is a well-known factorial:
\[
5040 = 7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1
\]

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Final Thoughts

This factorial connection makes 5040 a familiar and useful number in combinatorics, permutations, and divisibility.

Performing the Division

Now, divide 5040 by 24:

\[
5040 \div 24
\]

Rather than dividing directly, notice that dividing by 24 is the same as multiplying by its reciprocal, $ \frac{1}{24} $, but even better: simplify step-by-step:

\[
5040 \div 24 = (5040 \div 12) \div 2 = 420 \div 2 = 210
\]

Alternatively, break 24 down into factors — 24 = $6 \ imes 2 \ imes 2$, and since 5040 contains all the prime factors necessary to cancel these terms cleanly due to its factorial structure, the division resolves neatly.

The Mathematical Insight

This example highlights how complex denominators can be simplified behind the scenes through factorization. The expression:

\[
\frac{5040}{6 \cdot 2 \cdot 2}
\]