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?

Counting Frabjous Numbers

A 'frabjous' number is a 3-digit number where all digits are odd, and no adjacent digits are the same.

The odd digits are: 1, 3, 5, 7, 9. There are 5 odd digits available.

We need to determine the number of choices for each of the three positions (hundreds, tens, units).

- Hundreds digit: This digit must be odd. There are 5 choices (1, 3, 5, 7, 9).
- Tens digit: This digit must also be odd, but it cannot be the same as the hundreds digit. Since one odd digit is used for the hundreds place, there are $5-1=4$ choices remaining.
- Units digit: This digit must be odd and cannot be the same as the tens digit. It can be the same as the hundreds digit. Since one odd digit is used for the tens place, there are $5-1=4$ choices remaining.

Calculating Total Frabjous Numbers

To find the total number of possible frabjous numbers, we multiply the number of choices for each digit:

Total Numbers = (Choices for hundreds digit) $\times$ (Choices for tens digit) $\times$ (Choices for units digit)

Total Numbers = $5\times 4\times 4$

Total Numbers = $80$

Therefore, there are 80 such frabjous numbers.