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Understanding \frac{7!}{3! \cdot 2! \cdot 2!}: A Deep Dive into Factorials and Combinatorics
Understanding \frac{7!}{3! \cdot 2! \cdot 2!}: A Deep Dive into Factorials and Combinatorics
Factorials play a crucial role in combinatorics, probability, and algorithms across computer science and mathematics. One intriguing mathematical expression is:
\[
\frac{7!}{3! \cdot 2! \cdot 2!}
\]
Understanding the Context
This seemingly simple ratio unlocks deep connections to permutations, multiset arrangements, and efficient computation in discrete math. In this article, we’ll explore what this expression means, how to calculate it, and its significance in mathematics and real-world applications.
What Does \frac{7!}{3! \cdot 2! \cdot 2!} Represent?
This expression calculates the number of distinct permutations of a multiset — a collection of objects where some elements are repeated. Specifically:
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Key Insights
\[
\frac{7!}{3! \cdot 2! \cdot 2!}
\]
represents the number of unique ways to arrange 7 objects where:
- 3 objects are identical,
- 2 objects are identical,
- and another 2 objects are identical.
In contrast, if all 7 objects were distinct, there would be \(7!\) permutations. However, repeated elements reduce this number exponentially.
Step-by-Step Calculation
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Let’s compute the value step-by-step using factorial definitions:
\[
7! = 7 \ imes 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 5040
\]
\[
3! = 3 \ imes 2 \ imes 1 = 6
\]
\[
2! = 2 \ imes 1 = 2
\]
So,
\[
\frac{7!}{3! \cdot 2! \cdot 2!} = \frac{5040}{6 \cdot 2 \cdot 2} = \frac{5040}{24} = 210
\]
Thus,
\[
\frac{7!}{3! \cdot 2! \cdot 2!} = 210
\]
