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 only adjacent digits must differ, treat each position sequentially: first has $k$ choices, each next has $k-1$ choices (cannot repeat the immediate previous digit).

Answer 1

The Correct Option is D

Step 1: List allowed digits.
All digits must be odd $\Rightarrow$ allowed set $\{1,3,5,7,9\}$ (five choices).
Step 2: Choose the hundreds (first) digit.
Any of the 5 odd digits is allowed (a 3-digit number cannot start with $0$).
Choices for first digit $=5$.
Step 3: Choose the tens (second) digit with adjacency rule.
It must be odd and {different} from the first digit.
Choices for second digit $=5-1=4$.
Step 4: Choose the units (third) digit with adjacency rule.
It must be odd and {different} from the second digit (only adjacency matters).
Choices for third digit $=5-1=4$.

Step 5: Count the total numbers.
By multiplication principle: $5\times 4\times 4=\boxed{80}$. Final Answer:\fbox{80}