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. `^2P_5 2^5` b. `^7C_2 2^5` c. `^7C_2 5^2` d. none of these

Options

A. `""^(7)P_(5).2^(5)`
B. `""^(7)C_(2).2^(5)`
C. `""^(7)C_(2).5^(2)`
D. none of these

Explanation

Choose any two of the seven digits (in the seven digit number). This may be done in `""^(7)C_(2)` ways. Put 2 in these two digits. The remaining 5 digits may be arranged using 1 and 3 in `2^(5)` ways.
So, required number of numbers = `""^(7)C_(2)xx2^(5)`.