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: - Aurero
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:
\[
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]
Image Gallery
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
🔗 Related Articles You Might Like:
📰 Warning: CROSS AC Can Actually Do This—Proof Proven with Shocking Results! 📰 These 10 Girl Names Will Make You Fall Head Over Heels—You Won’t Believe Their Meanings! 📰 The Rarest Girl Names That’ll Steal Any Boy’s Heart Instantly—Shop Now! 📰 Why Every Home Needs A Circle Couch You Need This Stylish Centerpiece 📰 Why Every Home Needs A Felt Christmas Tree This Seasoncheck This Out 📰 Why Every Home Needs A Stunning Crystal Chandelier You Wont Believe The Difference 📰 Why Every Home Needs Checker Green Discover The Eye Catching Interior Trend Now 📰 Why Every Home Needs Old World Circa Lighting Heres Why Now 📰 Why Every Home Needs These Stunning Christmas Ornaments This Yearshop Before They Sell Out 📰 Why Every Interior Designer Is Obsessed The Celadon Color That Defies Trend Cycles 📰 Why Every Justice League Movie Cast Solemnly Backs Their Legacy Forever 📰 Why Every Kitchen Needs A Cereal Dispenser Youll Never Run Out Again 📰 Why Every Kitchen Needs A Chocolate Pudding Pieheres The Sweetest Reason Ever 📰 Why Every Kitchen Needs Chuck Steakfact Not Fiction Here 📰 Why Every Kitchen Needs This Must Have Chocolate Bread Recipewatch Sales Spike 📰 Why Every Man In Chess Should Invest In This Opinion Powering Mens Collection 📰 Why Every Minecraft Player Needs Cherry Blossom Biomes Believe Me Youll Regret Missing Them 📰 Why Every Modern Business Needs This Cute Cat Logo Click To Discover WhyFinal 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.
