A 'frabjous' number is defined as a 3-digit number with all digits odd, and no two adjacent digits being the same. For example, 135 is a frabjous number, while 133 is not. How many such frabjous numbers exist? A. 125 B. 729 C. 60 D. 80

GATE 2023 · General Aptitude · Number Theory · medium

Answer: There are 80 frabjous numbers. Answer: D.

1. Count choices for the hundreds digit (d1): d1 can be any of {1, 3, 5, 7, 9}. Number of choices = 5.
2. Count choices for the tens digit (d2): d2 must be an odd digit different from d1. With 5 odd digits total and 1 forbidden, choices = 4.
3. Count choices for the units digit (d3): d3 must be an odd digit different from d2. Choices = 4. Note: d3 can equal d1 (non-adjacent, so no constraint).
4. Apply multiplication principle: Total frabjous numbers = 5 * 4 * 4 = 80.