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

Answer 1

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

Other 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 ${}^7{C}_2$.

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