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 $2,5\text{and}7$. The smallest value of $n$ for which this is possible is

• A. $6$
• B. $9$
• C. $8$
• D. $7$

Answer 1

The correct option is D $7$

Given: $n-$digit numbers are to be formed using only $2,5\text{and}7$

$\therefore$ Each vacant space can be filled in $3$ ways $(2,5\text{and}7)$
We have to create $n-$digit number and each digit can be filled in three ways, therefore total number of such numbers are $3^n$
Total number of $n-$digit numbers should be $\ge 900$
$\Rightarrow 3^n\ge 900$
$\Rightarrow 3^6\le 900\le 3^7$
$\Rightarrow 729\le 900\le 2187$

Numbers should be greater than $900$, therefore the minimum value of $n$ should be $7.$