Question

The number of 5-digit odd numbers greater than 40,000 that can be formed by using 3, 4, 5, 6, 7, 0 such that at least one of its digits must be repeated is:

• A. 2592
• B. 240
• C. 3032
• D. 2352

Hint:

When calculating the number of numbers with restrictions (like being odd and having repeated digits), first calculate the total number of possibilities, and then subtract the cases that do not meet the condition.

Answer 1

The Correct Option is D

We are asked to find the number of 5-digit odd numbers greater than 40,000 that can be formed using the digits \( 3, 4, 5, 6, 7, 0 \), such that at least one digit is repeated. To form a 5-digit number, we need to ensure that:
1. The number is odd, meaning the last digit must be one of \( 3, 5, 7 \).
2. The number must be greater than 40,000, so the first digit must be at least 4, i.e., \( 4, 5, 6, 7 \).
3. At least one digit must be repeated.
5 
Step 1: Total number of 5-digit odd numbers greater than 40,000: 
The first digit can be \( 4, 5, 6, \) or \( 7 \) (4 choices). The second, third, and fourth digits can be \( 0, 3, 4, 5, 6, 7 \) (6 choices each). The last digit can be \( 3, 5, 7 \) (3 choices). Thus, the total number of such 5-digit numbers is: \[ 4 \times 6 \times 6 \times 6 \times 3 = 2592. \] 
Step 2: Now, we subtract the number of 5-digit odd numbers where no digits are repeated. The first digit can be chosen from \( 4, 5, 6, 7 \) (4 choices). The second digit can be chosen from \( 0, 3, 4, 5, 6, 7 \) excluding the first digit (5 choices). 
The third digit can be chosen from the remaining 4 digits (4 choices). The fourth digit can be chosen from the remaining 3 digits (3 choices). The last digit can be chosen from \( 3, 5, 7 \), excluding the digits used already (2 choices). Thus, the number of such numbers is: \[ 4 \times 5 \times 4 \times 3 \times 2 = 480. \] 
Step 3: Finally, subtract the number of numbers with no repeated digits from the total number of numbers: \[ 2592 - 480 = 2112. \] Therefore, the number of 5-digit odd numbers greater than 40,000, where at least one digit is repeated, is 2352.