\binom72 \times \binom92 = \left(\frac7 \times 62 \times 1\right) \times \left(\frac9 \times 82 \times 1\right) = 21 \times 36 = 756

Understanding the Combinatorial Equation: (\binom{7}{2} \ imes \binom{9}{2} = 756)

Factorials and combinations are fundamental tools in combinatorics and probability, helping us count arrangements and selections efficiently. One intriguing identity involves computing the product of two binomial coefficients and demonstrating its numerical value:

[
\binom{7}{2} \ imes \binom{9}{2} = \left(\frac{7 \ imes 6}{2 \ imes 1}\right) \ imes \left(\frac{9 \ imes 8}{2 \ imes 1}\right) = 21 \ imes 36 = 756
]

Understanding the Context

This article explores the meaning of this equation, how it’s derived, and why it matters in mathematics and real-world applications.

What Are Binomial Coefficients?

Before diving in, let’s clarify what binomial coefficients represent. The notation (\binom{n}{k}), read as "n choose k," represents the number of ways to choose (k) items from (n) items without regard to order:

$What is the value of $ n $ in the given equation? \[ \left(\frac{6 ...$

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$What is the value of $ n $ in the given equation? \[ \left(\frac{6 ...$

$The sum of first 10 terms of the series \( \left(x+\frac{1}{x}\right ...$

$integration - Find the integral $\int \frac{\left(\sin\left(2 \:x\right ...$

$f\left(\log \left(\frac{3}{2}\right)^{\frac{1}{2}}\right) & =4 e^{2 . \lo..$

Example 3. Simplify: (1681 )−43 ×[(925 )−23 ÷(25 )−3].Solution. (1681 )−..

Key Insights

[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
]

This formula counts combinations, a foundational concept in combinatorial mathematics.

Breaking Down the Equation Step-by-Step

We start with:

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Final Thoughts

[
\binom{7}{2} \ imes \binom{9}{2}
]

Using the definition:

[
\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \ imes 6}{2 \ imes 1} = \frac{42}{2} = 21
]

[
\binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \ imes 8}{2 \ imes 1} = \frac{72}{2} = 36
]

Multiplying these values:

[
21 \ imes 36 = 756
]

So,

[
\binom{7}{2} \ imes \binom{9}{2} = 756
]

Alternatively, directly combining expressions: