Question

The number of different seven digit numbers that can be written using only three digits 1, 2 &3 under the condition that the digit 2 occurs exactly twice in each number.
A. `""^(7)P_(5).2^(5)`
B. `""^(7)C_(2).2^(5)`
C. `""^(7)C_(2).5^(2)`
D. none of these

Answer 1

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)`.