Understand the formation rule

We can form $n$-digit numbers using only the digits $2, 5, 7$.

For each of the $n$ positions, there are exactly $3$ choices: $2,\ 5,\ 7$

Hence, the total number of distinct $n$-digit numbers is $3^n.$

Apply the condition

We need the smallest $n$ such that at least $900$ distinct numbers can be formed: $3^n \ge 900.$

Check powers of 3

$3^5 = 243$ $3^6 = 729$ $3^7 = 2187$

Now, $729 < 900 < 2187$

So the smallest value of $n$ satisfying $3^n \ge 900$ is $n=7.$

Match with options

Option $B$ is $7$.

Compare with stored correct answer

Stored correct answer: $B$

Our derived answer is also $B$.

Therefore, the stored answer is correct.