Q.3 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
A frabjous number is a 3-digit number where all digits are odd (1, 3, 5, 7, or 9) and no two adjacent digits are the same. There are 80 such numbers.
Calculation Method
Odd digits available: 1, 3, 5, 7, 9 (5 total).
- Hundreds place (first digit): 5 choices (any odd digit).
- Tens place (second digit): 4 choices (any odd digit except the first).
- Units place (third digit): 4 choices (any odd digit except the second).
Total: 5 × 4 × 4 = 80
Option Analysis
| Option | Calculation | Why Wrong? |
|---|---|---|
| (A) 125 | 5×5×5 | Counts all 3-digit numbers with odd digits, allowing adjacent repeats (5×5×5). Ignores “no two adjacent digits the same.” |
| (B) 720 | 10×9×8 | Matches permutations of 10 digits taken 3 at a time (10×9×8), but ignores odd digits only and 3-digit constraint. |
| (C) 60 | 5×4×3 | Assumes all digits distinct (5×4×3), but third digit can repeat first if not adjacent to second. Over-restricts condition. |
| (D) 80 | 5×4×4 | Correct, as verified by position choices and code enumeration. |
What Defines a Frabjous Number?
A frabjous number meets two rules:
- 3 digits, from 100-999.
- All digits odd: only 1, 3, 5, 7, 9 (5 options).
- No adjacent repeats (e.g., 137 yes; 133 no).
Step-by-Step Frabjous Number Calculation
- Hundreds digit: 5 choices (1,3,5,7,9). Can’t be 0 anyway.
- Tens digit: 4 choices (odd digits minus hundreds digit).
- Units digit: 4 choices (odd digits minus tens digit; can match hundreds).
Formula: 5×4×4=80
Python Proof (enumerates all)
odd_digits = ['1','3','5','7','9']
count = 0
for h in odd_digits:
for t in odd_digits:
if t != h:
for u in odd_digits:
if u != t:
count += 1
print(count) # Output: 80
Exam Tips for Frabjous Number Questions
- Focus on positional restrictions for adjacent rules.
- Total odd 3-digit numbers: 125; subtract adjacent repeats for verification.
- Practice permutations with constraints for CSIR NET/GATE quant sections.
Master 3-digit odd digits no adjacent same problems with this method—answer is always 80!
