Question

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

• A. ${}^7{P}_2\times 2^5$
• B. ${}^7{C}_2\times 2^5$
• C. ${}^7{C}_2\times 5^2$
• D. None of these

Answer 1

The correct option is B ${}^7{C}_2\times 2^5$

Other than $2$, 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 ${}^7{C}_2$.
Hence, the required number of ways is $={}^7{C}_2\times 2^5$.