Question

A 'frabjous' number is defined as a 3-digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

• A. 125
• B. 720
• C. 60
• D. 80

Hint:

When dealing with digit problems with restrictions, always multiply choices step by step, considering restrictions on adjacency or repetition.

Answer 1

The Correct Option is D

Step 1: Define the set of digits.
The odd digits are: \(\{1, 3, 5, 7, 9\}\). So there are 5 choices for each digit if considered independently.

Step 2: Condition for the first digit.
The number must be a 3-digit number. The first digit can be any odd digit (5 choices).

Step 3: Condition for the second digit.
The second digit must be odd, but not equal to the first digit. So there are \(5 - 1 = 4\) choices.

Step 4: Condition for the third digit.
The third digit must be odd, but not equal to the second digit. Again, there are \(4\) choices.

Step 5: Total count.
\[ \text{Total frabjous numbers} = 5 \times 4 \times 4 = 80 \]

Final Answer: \[ \boxed{80} \]