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  • Determine the number of distinct six-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6, allowing repetition, where exactly two digits are odd.

Determine the number of distinct six-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6, allowing repetition, where exactly two digits are odd.

 

Option: 1

7895

 


Option: 2

6045

 


Option: 3

3402

 


Option: 4

1249


Answers (1)

best_answer

To determine the number of distinct six-digit numbers that can be formed using the digits 1,2,3,4, 5 , and 6 , allowing repetition, where exactly two digits are odd, we can consider the different cases.

Case 1: Two odd digits and four even digits:

There are 6 choices for the positions of the odd digits and 3 choices for each odd digit. The remaining positions can be filled with any of the even digits, which are 2, 4, and 6 . Therefore, there 6 \times 3 \times 3 \times 3 \times 3 \times 3=1458 possible arrangements in this case.

Case 2: One odd digit repeated twice and four even digits:

There are 6 choices for the position of the repeated odd digit and 3 choices for the odd digit. The remaining positions can be filled with any of the even digits, resulting in 3 choices. Therefore, there are 6 \times 3 \times 3 \times 3 \times 3 \times 1=486 possible arrangements in this case.

Case 3: Four odd digits and two even digits:

There are 6 choices for the positions of the even digits and 3 choices for each even digit. The remaining positions can be filled with any of the odd digits, which are 1,3 , and 5 . Therefore, there are 6 \times 3 \times 3 \times 3 \times 3 \times 3=1458 possible arrangements in this case.

Therefore, the total number of distinct six-digit numbers that can be formed, where exactly two digits are odd, is

1458+486+1458=3402.

It is important to note that repetition is allowed in this case, as the digits 1,2,3,4,5, and 6 can be used multiple times.

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