How many ternary strings that is strings made up of 0s 1s and 2s of length 5 contain at most two 0s at most two 1s and at most two 2s?

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There are 3 basic categories here, as there has to be at least one of one number, and two of the other two numbers. So we have:

$\text{01122}$
$\text{10022}$
$\text{20011}$

So I take each these and permutate. Which gives me $3 \times 5!$ which is wrong. The answer is $90$. I know it is wrong because $3 \times 5! > 3^5$ which shouldn't be possible. I can't find any intuitive reason why my answer is wrong.
How did I approach it wrong?