Solution: We compute the number of distinct permutations of 10 sensors: 4 red (R), 5 green (G), and 1 blue (B). The number of sequences is: - Sourci
Solution: How to Compute Distinct Permutations of 10 Sensors with Repeated Colors
Solution: How to Compute Distinct Permutations of 10 Sensors with Repeated Colors
When designing systems involving sequences of objects—like arranging colored sensors—understanding the number of distinct arrangements is crucial for analysis, scheduling, or resource allocation. In this problem, we explore how to calculate the number of unique permutations of 10 sensors consisting of 4 red (R), 5 green (G), and 1 blue (B).
The Challenge: Counting Distinct Permutations with Repetitions
Understanding the Context
If all 10 sensors were unique, the total arrangements would be \(10!\). However, since sensors of the same color are indistinguishable, swapping two red sensors does not create a new unique sequence. This repetition reduces the total number of distinct permutations.
To account for repeated elements, we use a well-known formula in combinatorics:
If we have \(n\) total items with repeated categories of sizes \(n_1, n_2, ..., n_k\), where each group consists of identical elements, the number of distinct permutations is given by:
\[
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]
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Key Insights
Applying the Formula to Our Sensor Problem
For the 10 sensors:
- Total sensors, \(n = 10\)
- 4 red sensors → \(n_R = 4\)
- 5 green sensors → \(n_G = 5\)
- 1 blue sensor → \(n_B = 1\)
Plug into the formula:
\[
\ ext{Number of distinct sequences} = \frac{10!}{4! \cdot 5! \cdot 1!}
\]
Step-by-step Calculation
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Compute factorials:
\(10! = 3628800\)
\(4! = 24\)
\(5! = 120\)
\(1! = 1\) -
Plug in:
\[
\frac{3628800}{24 \cdot 120 \cdot 1} = \frac{3628800}{2880}
\]
- Perform division:
\[
\frac{3628800}{2880} = 1260
\]
Final Answer
There are 1,260 distinct permutations of the 10 sensors (4 red, 5 green, and 1 blue).
Why This Matters
Accurately calculating distinct permutations helps in probability modeling, error analysis in manufacturing, logistical planning, and algorithmic design. This method applies broadly whenever symmetries or redundancies reduce the effective number of unique arrangements in a sequence.
