Answer: Option B
:
B
The total number of numbers between 10 and 1000 are 989 but we have to form the numbers by using numerals 1,2,........9,so the numbers containing any '0' would be excluded i.e., Required number of ways
= 989 - $\left\{\begin{array}{c}20,30,40,...........................100=9\\ 101,102,...........................200=19\\ 201,...............................300=19\\ .........................................\\ .........................................\\ 901,..............................900=18\end{array}\right\}$
= $989-(9+18+19\times 8)$
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 = $989-(9+18+19\times 8)$
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.