Question

If \[n\in N\] and $300 < n < 3000$ and ‘n’ is made of digits by taking from 0, 1, 2, 3, 4, 5 then the greatest possible number of values of ‘n’ is (repetition is allowed)?
1.539
2.260
3.320
4.300

Answer 1

Hint: In order to answer the question, first we made two cases; first case is of three digit number and second case is of four digit number. We then count the numbers of 3-digit numbers and the number of 4-digit numbers and hence we get the required answer. We need to remember the condition i.e. \[n\in N\] and $300 < n < 3000$.

Complete step by step solution:
We have given that,
\[\Rightarrow\] 300Case 1:
Let suppose we have formed 3 digit number more than 300,
Only 3 or 4 or 5 can be the first digit, therefore 3 possibilities for the first digit (3 ways).
Second digit can be filled with any of the 6 digits given, therefore 6 possibilities are there.
Third digit again can be filled with any of the 6 digits given, therefore 6 possibilities or 6 ways are there.
Thus, total count = \[3\times 6\times 6=108\]
We have to subtract 1 number from the total of 108 numbers as the number 300 cannot be included.
Therefore, total count = 108-1=107
\[\therefore number\ of\ 3digit\ number=107\]
Case 2:
If we formed 4-digit number which is less than 3000,
We can pick only 1 or 2 in the first place of the 4-digit number, so there are 2 ways for the first place.
Second digit can be filled with any of the 6 digits given, therefore 6 possibilities are there.
Third digit again can be filled with any of the 6 digits given, therefore 6 possibilities or 6 ways are there.
Fourth digit again can be filled with any of the 6 digits given, therefore 6 possibilities or 6 ways are there.
Therefore, total count= \[2\times 6\times 6\times 6=432\]
\[\therefore number\ of\ 4\ digit\ number=432\]
Hence,
\[\Rightarrow n=432+107=539\]
Therefore, there are a total of 539 numbers that can be formed by taking the digits from 0, 1, 2, 3, 4, 5 and \[n\in N\] and $300 < n < 3000$.

\[\therefore option(1)\ \] is the correct answer.

Note:
In solving these types of questions, we have to keep in mind the condition that is given in the question because sometimes students solve the whole solution correctly and just by ignoring the condition, they will get the wrong answer. For example; in the above question the condition is \[n\in N\] and 300 < n < 3000, so when we find the total count of 3-digit numbers we have to subtract 1 from the total count because it satisfies the required condition.