\binom72 \times \binom92 = \left(\frac7 \times 62 \times 1\right) \times \left(\frac9 \times 82 \times 1\right) = 21 \times 36 = 756 - Crosslake
Understanding the Combinatorial Equation: (\binom{7}{2} \ imes \binom{9}{2} = 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:
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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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[
\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:
[
\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
]
