Question

$n-$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. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

The correct option is B $7$

Consider how many digits you have to choose at each spot: for the one's spot, you have 3 digits; for the tens spot, you have three digits; all the way to the nth spot, you have 3 digits to choose from. As such, the total number of ways to make an n-digit number with only $2,5$ and $7$ is going to be $3^n$.

Thus we're looking for the smallest integer $n$ such that $3^n\ge 900$,

or in other words, since the $\log$ function is increasing, $n\ln \left(3\right)\ge ln\left(900\right)$

on dividing by $\ln 3$,

we get, $n\ge 6.19$ Thus, $n=7$