Simplifying the Factorial Expression: Understanding \frac{13!}{6! \cdot 4! \cdot 3!}

Factorials can seem intimidating at first, but expressions like \(\frac{13!}{6! \cdot 4! \cdot 3!}\) offer a fascinating glimpse into combinatorial mathematics and probability. This article breaks down the meaning, calculation, and real-world significance of this particular factorial fraction.

Understanding the Context

What Is \(\frac{13!}{6! \cdot 4! \cdot 3!}\)?

The expression \(\frac{13!}{6! \cdot 4! \cdot 3!}\) arises from combinatorics and is commonly related to generalized multinomial coefficients. While it doesn’t directly represent a standard binomial coefficient, it reflects a product of factorials divided across multiple groups — useful in complexity-intensive calculations.

Step-by-Step Simplification

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Key Insights

Start with the definition of 13 factorial:

\[
13! = 13 \ imes 12 \ imes 11 \ imes 10 \ imes 9 \ imes 8 \ imes 7 \ imes 6!
\]

Notice that \(6!\) appears in the numerator, so it cancels with the \(6!\) in the denominator:

\[
\frac{13!}{6!} = 13 \ imes 12 \ imes 11 \ imes 10 \ imes 9 \ imes 8 \ imes 7
\]

Now divide by \(4! \cdot 3! = (4 \ imes 3 \ imes 2 \ imes 1) \cdot (3 \ imes 2 \ imes 1) = 24 \ imes 6 = 144\):

Continue Reading

r = \frac{a + b - t}{2}
\frac{\pi r^2}{\frac{1}{2}ab} = \frac{2\pi r^2}{ab}
Substituting $r = \frac{a + b - t}{2}$:

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Final Thoughts

So,

\[
\frac{13!}{6! \cdot 4! \cdot 3!} = \frac{13 \ imes 12 \ imes 11 \ imes 10 \ imes 9 \ imes 8 \ imes 7}{144}
\]

Calculate numerator:

\[
13 \ imes 12 = 156 \
156 \ imes 11 = 1716 \
1716 \ imes 10 = 17160 \
17160 \ imes 9 = 154440 \
154440 \ imes 8 = 1235520 \
1235520 \ imes 7 = 8648640
\]

Now divide:

\[
\frac{8648640}{144} = 60060
\]

Final Result

\[
\frac{13!}{6! \cdot 4! \cdot 3!} = 60060
\]