To find the smallest number with distinct digits whose digits add up to $45$, we need to consider the given options and determine which one satisfies the conditions.
Let's analyze the options provided:
- $123555789:$ The sum of the digits in this number is $1+2+3+5+5+5+7+8+9 = 45$. However, this number contains repeating digits (three $5$s), which contradicts the requirement of distinct digits.
- $123457869:$ The sum of the digits in this number is $1+2+3+4+5+7+8+6+9 = 45$. Additionally, all the digits in this number are distinct, which satisfies the condition of distinct digits. However, we need to find the smallest number that meets the criteria.
- $123456789:$ The sum of the digits in this number is $1+2+3+4+5+6+7+8+9 = 45$. Moreover, all the digits in this number are distinct. Importantly, this is the smallest arrangement of ${\color{Red}\text{distinct digits}}$ that adds up to $45$. Each digit from $1$ to $9$ is used exactly once, and there is no smaller number with distinct digits that satisfies the condition.
- $99999:$ The sum of the digits in this number is $9+9+9+9+9 = 45$. However, this number contains repeated digits (five $9$s), which does not satisfy the requirement of distinct digits.
Based on the analysis, the correct answer is $123456789$. It is the smallest number with distinct digits whose digits add up to $45$.
$\textbf{Short Method: }$ Options A and D are not valid choices as they both have repeated digits. On the other hand, when comparing options B and C, it becomes evident that $123456789$ is indeed the smallest number with distinct digits whose digits add up to $45$.