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. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

Answer: (B)

$7$

We have $3$ numbers $2,5,\xi 7.$

For units place we can use any of the $3,$

For tens place we can use any of the $3,$ and so on until $nth$ place.

Total permutations $=3\times 3\times ......n$ times$=$ Total number of numbers.

$\Rightarrow 3^n\ge 900$

$\Rightarrow n\log 3\ge \log 900$

$\Rightarrow n\ge 6.27$

$\Rightarrow n=7$

Hence, the answer is $7.$