An interesting AMC 10 probability problem

Problem 20 from the 2006 AMC 10 gave my son some trouble last night:

Problem #20 from the 2006 AMC 10 A

Here’s the problem:

Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?

I like this problem because it seems overwhelming at first, but a little exploration reveals the hidden structure. The lessons learned working through this problem seemed important enough to revisit again this morning.

Here’s the introduction to the problem and a few test cases:


At the end of the last video we started talking about the Pigeon Hole principle – a simple sounding, but really powerful idea in math.


In the last part of our talk today I introduced the idea of looking at the remainder when you divide by 5. This idea uses “pigeon holes” that are slightly easier to see (but only slightly), but it gave a chance to look at the problem one more time through a different lens:


So, a fun and really instructive problem. As I think about it, I’d love to see how other people would approach this problem with younger math students.

One thought on “An interesting AMC 10 probability problem

  1. Going in the opposite direction, what if you just choose 2 numbers (or 3, 4, 5)? Then what is the probability that two will differ by a multiple of 5?

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