An $n$-digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$-digit numbers are to be formed using only the three digits $2, 5$ and $7$. The smallest value of $n$ for which this is possible is

How many ways are there to arrange the letters of the word $EDUCATION$ so that all the following three conditions hold?
- The vowels occur in the same order $(E, U, A, I, O)$.
- The consonants occur in the same order $(D, C, T, N)$.
- No two consonants are next to each other.

How many words of $4$ distinct letters can be formed using the letters of the word $DHOLPUR$ if $L$ and $P$ are always excluded?

The number of different $9$-digit numbers that can be formed by rearranging the digits of the number $445577888$ such that the odd digits occupy even positions is:

If all the words with or without meaning made using all the letters of the word $NAGPUR$ are arranged as in a dictionary,then the word at $315^{\text{th}}$ position in this arrangement is:

The sum of all $4$-digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is