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 $2,5$ & $7$. The smallest value of $n$ for which this is possible is:

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

Answer: (B)

$7$

Distinct $n-digit$ numbers which can be formed using digits $2,5$ and $8$ are $3^n$. We have to find $n$ so that

$3^n\ge 900$

$\Rightarrow 3^n{-}^2\ge 100$

$\Rightarrow n-2\ge 5$

$\Rightarrow n\ge 7$

So the least value of $n$ is $7$.