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:

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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Final Thoughts

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.