\( 2(^{4}P_3) \)
We are given the digits \( \{3, 5, 6, 7, 8\} \) and need to form 4-digit numbers between 6000 and 9999 without repetition.
Step 1: Identify Valid Leading Digits
- A number must be \(\geq 6000\), so the first digit must be 6, 7, or 8 (valid leading digits). - The remaining 3 digits are selected from the remaining 4 digits.
Step 2: Count Even and Odd Numbers
- A number is even if its last digit is even (\(6,8\)). - A number is odd if its last digit is odd (\(3,5,7\)).
Step 3: Count Odd and Even Cases
1. Case 1: Last digit is even (6 or 8) - First digit choices: \( 6, 7, 8 \) (3 choices). - Last digit choices: \( 6 \) or \( 8 \) (2 choices). - Middle two digits are chosen from remaining 3 digits: \( ^3P_2 \). \[ \text{Total even numbers} = 3 \times 2 \times {}^3P_2. \] 2. Case 2: Last digit is odd (3,5,7) - First digit choices: \( 6, 7, 8 \) (3 choices). - Last digit choices: \( 3,5,7 \) (3 choices). - Middle two digits are chosen from remaining 3 digits: \( ^3P_2 \). \[ \text{Total odd numbers} = 3 \times 3 \times {}^3P_2. \]
Step 4: Compute Difference Between Odd and Even Numbers
\[ \text{Difference} = (3 \times 3 \times {}^3P_2) - (3 \times 2 \times {}^3P_2). \] \[ = 3(3 - 2) {}^3P_2 = 3 {}^3P_2 = {}^4P_3. \]
Step 5: Conclusion
Thus, the difference between the number of odd and even numbers is: \[ \boxed{^{4}P_3}. \]
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