How many ten-digit numbers are there in which every digit is 2 or 3, and no two 3s are adjacent?
Taken from the 2008 IMC https://chiuchang.org/wp-content/uploads/sites/2/2018/02/2008-IWYMIC-Individual.x17381.pdf
my attempt
The number of ten digit numbers in which the digits are either 2 or 3 is $2^{10}$ and the numbers of ten digit numbers where there is no pair of adjacent numbers that are the same is $2$ E.g($2323232323$ & $3232323232$) and we know that the number of ten digit numbers consisting of pairs of adjacent $2$s and adjacent $3$s is the same due to symmetry therefore the answer would be $\frac{2^{10}-2}{2}=511$ however this doesn't taken into account the possibilities of having adjacent pairs of $3$s and $2$s in the same number.