Solution: The total items are $5 + 7 = 12$, with 5 red and 7 blue. The number of distinct sequences is $\frac{12!}{5!7!} = 792$. Therefore, the answer is $\boxed{792}$.Question: A robotic engineer is programming a machine to assemble components using exactly 3 different colored

["Solution to Color-Based Component Assembly: Counting Distinct Sequences", "When programming automated assembly systems, robotic engineers often face combinatorial challenges—especially when assembling parts in a specific sequence based on color. Consider a scenario where a machine must arrange exactly 3 colored components using two distinct colors: red and blue. Suppose the task requires using a total of 5 red components and 7 blue components. How many unique sequences can the robot execute?", "### The Problem in Numbers\nWe are tasked with arranging 5 red (R) and 7 blue (B) components into a full sequence of 12 parts. Since the components of the same color are indistinguishable, the challenge lies in counting how many distinct orderings exist.", "This is a classic multinomial coefficient problem. The total number of ways to arrange 12 items where 5 are identical of one kind and 7 are identical of another is given by:", "[\n\frac{12!}{5! \cdot 7!}\n]", "### Why This Formula Works\n- 12! represents the total permutations if all components were unique.\n- Because 5 red components are identical, swapping any two red parts produces the same visual sequence — reducing the count by a factor of 5!.\n- Similarly, the 7 blue components are indistinguishable among themselves, so their internal swaps reduce the arrangements by 7!.\n- Dividing total permutations by factorial counterparts yields only the distinct sequences.", "### Calculation Breakdown\nFirst, compute:\n[\n12! = 479001600\n]\n[\n5! = 120, \quad 7! = 5040\n]\n[\n5! \cdot 7! = 120 \cdot 5040 = 604800\n]\nNow divide:\n[\n\frac{12!}{5! \cdot 7!} = \frac{479001600}{604800} = 792\n]", "### Final Answer\nThus, the number of distinct sequences in which the robot can assemble 5 red and 7 blue components is:\n[\n\boxed{792}\n]", "This mathematical foundation ensures precision in automation programming, enabling efficient, error-free production lines where component identity and sequence matter."]