Question

The number of different seven-digit numbers that can be written using only the digits 1, 2, and 3 with the condition that the digit 2 occurs twice in each number is:

• A. \( ^7C_2 \times 5 \)
• B. \( ^7P_2 \times 5 \)
• C. \( ^7C_2 \times 2 \)
• D. None of these

Hint:

When forming numbers with repetitions of certain digits, start by determining how to place the repeated digits, and then consider how to fill the remaining positions.

Answer 1

The Correct Option is A

Others than 2, the remaining five places can be filled by 1 and 3 for each place. 
The number of ways for five places is \(2 \times 2 \times 2 \times 2 \times 2 = 2^5\). For 2, selecting 2 places out of 7 is \(^7C_2\). 
Hence, the required number of ways is \(^7C_2 \times 2^5\). 
Therefore, the correct answer is Option A.