To determine how many positive \(10\)-digit numbers are cozy, we must first analyze the definition of a cozy number. A number is considered cozy if every digit in the number is either \(3\) or is next to a \(3\). Thus, the valid digits for a cozy number can be summarized as follows:
- The digit \(3\) itself.
- The digits \(2\) and \(4\) (since they are adjacent to \(3\)).
- However, digits \(0, 1, 5, 6, 7, 8, 9\) are not allowed since they are not adjacent to \(3\).
This means that the only digits that can appear in a cozy number are \(2\), \(3\), and \(4\).
First Digit Constraint
Since we are forming \(10\)-digit numbers, the very first digit (the most significant digit) cannot be \(0\). In this particular case, as none of our allowed digits include \(0\), the first digit can be \(2\), \(3\), or \(4\).
Total Choices Calculation
For each of the remaining \(9\) digits, they can independently be either \(2\), \(3\), or \(4\). Therefore, here's the breakdown:
- First digit: It can be \(2\), \(3\), or \(4\) (3 options).
- Remaining 9 digits: Each can also be \(2\), \(3\), or \(4\) (3 options for each of the 9 digits).
Putting this together, we calculate the total number of cozy \(10\)-digit numbers:
\[ \text{Total cozy } 10\text{-digit numbers} = \text{Choices for first digit} \times \text{Choices for remaining 9 digits} \]
We have:
\[ \text{Total cozy } 10\text{-digit numbers} = 3 \times (3^9) \]
Calculating \(3^9\):
\[ 3^9 = 19683 \]
Thus, the total becomes:
\[ 3 \times 19683 = 59049 \]
Conclusion
Therefore, the total number of positive \(10\)-digit cozy numbers is:
\[ \boxed{59049} \]
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