• # question_answer How many numbers lying between 10 and 1000 can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition is allowed) A) 1024 B) 810 C) 2346 D) None of these

The total number between 10 and 1000 are 989 but we have to form the numbers by using numerals $1,\ 2,........9,\ \ i.e.$0 is not occuring so the numbers containing any ?0? would be excluded i.e., Required number of ways =989-\left\{ \begin{align} & 20,\ 30,\ 40,................100=9 \\ & 101,\ 102,...................200=19 \\ & 201,........................300=19 \\ & ............................................. \\ & ............................................. \\ & 901,..........................990=18 \\ \end{align} \right\} $=989-(9+18+19\times 8)=810$. Aliter: Between 10 and 1000, the numbers are of 2 digits and 3 digits. Since repetition is allowed, so each digit can be filled in 9 ways. Therefore number of 2 digit numbers $=9\times 9=81$ and number of 3 digit numbers $9\times 9\times 9=729$ Hence total ways$=81+729=810$.