Question

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

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$

Using $2,5$ and $7$ with repetition each place of $n$ digit number can be chosen in $3$ ways.

Hence, total number of $n-$digit numbers $=3\times 3\times \mathrm{3...}n$ times $=3^n.$

According to given condition $3^n\ge 900\Rightarrow 3^{n-2}\ge 100$

$\therefore n-2\ge 5\Rightarrow n\ge 7$