To determine the number of ${\color{Red}\text{“frabjous”}}$ numbers, we consider the following conditions:
- The number must have three digits.
- All digits must be odd.
- No two adjacent digits can be the same.
For the first digit, we have five odd choices: $1, 3, 5, 7,$ and $9.$
Once we select the first digit, there are four remaining odd digits available for the second digit. We exclude the digit chosen for the first digit and its adjacent digit, as they would violate the condition of no adjacent identical digits.
For the third digit, we have four remaining odd digits to choose from. We exclude the digit chosen for the second digit and its adjacent digit and include the first digit.
Hence, the total number of frabjous numbers is found by multiplying the choices for each digit: $5\; ($choices for the first digit$) \times\; 4\; ($choices for the second digit$)\; \times 4\; ($choices for the third digit$) = 80.$
$\textbf{Short Method:}$ We have total five numbers: $1, 3, 5, 7,$ and $9.$
$$\underbrace{\_\_\_\_{\color{Red} {\boxed{1}}}\_\_\_\_}_{\underbrace{\text{Choose anything from 1, 3, 5, 7, 9}}_{\text{5 choices}}} \quad \underbrace{\_\_\_\_{\color{Purple} {\boxed{3}}}\_\_\_\_}_{\underbrace{\text{Choose 3, 5, 7, 9 but not 1}}_\text{4 choices}} \quad \underbrace{\_\_\_\_{\color{Red} {\boxed{1}}}\_\_\_\_}_{\underbrace{\text{Choose 1, 5, 7, 9 but not 3}}_\text{4 choices}} $$
Correct Answer: D
