我想学习一下这个数学题
The Gaussbot factory assembles robots. Each robot comes in one of three colours:
red, blue, or green. Each robot also has a number stamped on its head: 1, 2, 3, or 4.
The nth robot assembled is the first robot to have the same colour and the same
number as a previously assembled robot. What is the greatest possible value of n?
(A) 11 (B) 12 (C) 13 (D) 7 (E) 8
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Let's understand this problem step by step. The Gaussbot factory assembles robots, where each robot has two attributes: a color and a number. There are 3 possible colors - red, blue, and green - and 4 possible numbers from 1 to 4. We need to find the greatest possible value of n, where the nth robot is the first one to have the same color and number combination as a previously assembled robot.
Now let's count how many distinct robot types are possible. Since each robot has a color and a number, we can think of this as a grid. We have 3 colors in rows and 4 numbers in columns. Each cell in this grid represents a unique robot type. By multiplying 3 colors times 4 numbers, we get 12 distinct robot types total. This is a crucial insight for solving the problem.
This is where the Pigeonhole Principle comes in. This fundamental principle states that if you have more items than containers, at least one container must contain more than one item. In our robot problem, the 12 distinct robot types act as pigeonholes, and the assembled robots are the pigeons. If we try to place 13 pigeons into 12 holes, at least one hole must contain two pigeons - meaning we must have a repeated robot type.
Now let's determine the maximum possible value of n. In the best case scenario, the first 12 robots are all distinct types, using up all possible combinations. The 13th robot must then repeat one of the previous types, making it the first repetition. Since n represents the position of the first repeated robot, the maximum value of n is 13. This occurs when we delay the first repetition as long as possible by making the first 12 robots all different.
Let's conclude our solution. We determined that there are 12 distinct robot types possible. Using the Pigeonhole Principle, if we assemble 13 robots, at least one type must be repeated. The maximum value of n occurs when we delay this repetition as long as possible - by making the first 12 robots all distinct types. Therefore, the 13th robot is guaranteed to be the first repetition, giving us n equals 13. The correct answer is choice C: 13.