In secondary school, students may come across combinations and permutations as part of a course in statistics. In mathematics, these are technical terms with precise meanings and formulas. However, there are many problems that are not immediately solvable using just these types of arrangements. What is more common is that you’ll get a question that requires enumerating all possible sets that fit some criteria and then figuring out how many arrangements there are of the elements of each set. Here is one example of such a question.

**The Question**A message is written with only the binary digits 1 and 0. A valid message can contain at most three consecutive 1s or 0s. For example, 11000101 and 10101010 are valid but 01111010 is not. Given that a message has a fixed length of just ten binary digits, how many valid messages can be created?

As always, I need both the final answer and a valid method.

Good Luck!

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**This PMQ11 competition closes on MONDAY 25 March at 23:59 GMT - one extra weekday from now on**.

**The Prize****The prizes for this PMQ are 3 free places in our Online Maths Club for 3 months.**The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

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Enjoy the challenge!

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