The number of different seven-digit numbers that can be written...

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 ${ }^7 C_2 \times 2^5$.