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Age 11 to 14 Short
Challenge Level

The digits $1$ and $3$ will always be followed by the digit $2$. The digit $2$ can be followed by either $1$ or $3$. Hence the digit $2$ appears exactly five times in a ten digit number, in alternate positions.
 
If the first digit is $2$, then in each even position we have two choices, $1$ or $3$. This gives $2\times 2\times 2\times 2\times 2 = 32$ possibilities. Otherwise, the second digit is $2$ and in each odd position we have two choices. So again there are $32$ possibilities, making a total of $64$
 
 
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.