Question

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

Answer 1

We need to find $n-$digit numbers using $2,5$ & $7.$ Each digit can be any of the $3$ numbers.

Total numbers possible $=3\times 3\times \mathrm{3.....}n$ times

$=3^n$

Given there must be at least $900.$

$\Rightarrow 3^n\ge 900$

$\Rightarrow n\log 3\ge 9\left(\log 9\right)+2$

$\Rightarrow (n-2)0.447\ge 2$

$\Rightarrow n\ge 6.2$

$\Rightarrow n=7.$

Hence, the answer is $7.$