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:

For counting problems with restrictions like "not adjacent," handle the positions sequentially. Calculate the choices for the first position, then use that to determine the restricted choices for the second, and so on.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves permutations and combinations, specifically using the multiplication principle of counting to find the number of possible 3-digit numbers that satisfy a given set of conditions.
Step 2: Key Formula or Approach:
We will use the multiplication principle. If an event can occur in \(m\) ways, and a second event can occur in \(n\) ways, then the two events can occur in sequence in \(m \times n\) ways. We will determine the number of choices for each of the three digits (hundreds, tens, and units) based on the given rules.
Step 3: Detailed Calculation:
The conditions for a 3-digit number to be 'frabjous' are:
1. All three digits must be odd.
2. No two adjacent digits can be the same.
The set of odd digits is \{1, 3, 5, 7, 9\}. There are 5 odd digits.
Let the 3-digit number be represented by three places: H (Hundreds), T (Tens), U (Units).
Choices for the Hundreds place (H):
Any of the 5 odd digits can be chosen.
Number of choices for H = 5.
Choices for the Tens place (T):
This digit must be odd, but it cannot be the same as the digit in the Hundreds place.
So, we have 5 odd digits minus the 1 digit already used for H.
Number of choices for T = 5 - 1 = 4.
Choices for the Units place (U):
This digit must be odd, but it cannot be the same as the digit in the adjacent Tens place. It can, however, be the same as the digit in the Hundreds place.
So, we have 5 odd digits minus the 1 digit already used for T.
Number of choices for U = 5 - 1 = 4.
Total number of frabjous numbers:
Using the multiplication principle, the total number of ways is the product of the number of choices for each place.
\[ \text{Total Numbers} = (\text{Choices for H}) \times (\text{Choices for T}) \times (\text{Choices for U}) \] \[ \text{Total Numbers} = 5 \times 4 \times 4 = 80 \] Step 4: Final Answer:
There are 80 such frabjous numbers.
Step 5: Why This is Correct:
The calculation correctly applies the given constraints. There are 5 options for the first digit, and for each subsequent digit, there are 4 options (any odd digit except the one immediately preceding it). Therefore, 5 * 4 * 4 = 80 is the correct total.