Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Rule): If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both. This extends to any number of independent events.
• $n$-digit numbers with repetition: If we are forming $n$-digit numbers using $k$ distinct digits, and repetition is allowed, then the total number of distinct numbers that can be formed is $k^n$.
• Inequalities: Understanding how to set up and solve inequalities to find the minimum or maximum value that satisfies a given condition.

Step-by-Step Solution

Step 1: Identify the parameters and the goal.

We are given that we need to form $n$-digit numbers using only the digits 2, 5, and 7. This means we have 3 choices for each digit. Repetition is allowed. Our goal is to find the smallest value of $n$ such that we can form at least 900 distinct numbers.

Step 2: Determine the number of $n$-digit numbers that can be formed.

Since we have 3 choices for each of the $n$ digits, and repetition is allowed, the total number of $n$-digit numbers that can be formed is $3^n$. This follows directly from the Fundamental Principle of Counting. $\text{Total number of distinct numbers} = 3^n$

Step 3: Set up the inequality.

We want to find the smallest $n$ such that the number of distinct numbers formed is greater than or equal to 900. This translates to the inequality: $3^n \ge 900$

Step 4: Solve the inequality by testing integer values of n.

We need to find the smallest integer $n$ that satisfies the inequality. We test increasing values of $n$:

• For $n = 1$: $3^1 = 3$. Since $3 < 900$, $n=1$ is not a solution.
• For $n = 2$: $3^2 = 9$. Since $9 < 900$, $n=2$ is not a solution.
• For $n = 3$: $3^3 = 27$. Since $27 < 900$, $n=3$ is not a solution.
• For $n = 4$: $3^4 = 81$. Since $81 < 900$, $n=4$ is not a solution.
• For $n = 5$: $3^5 = 243$. Since $243 < 900$, $n=5$ is not a solution.
• For $n = 6$: $3^6 = 729$. Since $729 < 900$, $n=6$ is not a solution.
• For $n = 7$: $3^7 = 2187$. Since $2187 \ge 900$, $n=7$ is a solution.

Since $n=6$ is too small, and $n=7$ is the next smallest integer, $n=7$ is the smallest integer that satisfies the inequality.

Common Mistakes & Tips

• Forgetting Repetition: Failing to recognize that repetition is allowed would lead to a completely different (and incorrect) approach.
• Incorrect Inequality: Setting up the inequality as $3^n = 900$ would be incorrect since we are looking for the smallest $n$ for which at least 900 numbers can be formed.
• Miscalculation: Errors in calculating the powers of 3 could lead to selecting the wrong value of $n$.

Summary

The problem involves finding the smallest number of digits, $n$, such that at least 900 distinct numbers can be formed using the digits 2, 5, and 7 with repetition allowed. This is solved by recognizing that the number of distinct $n$-digit numbers is $3^n$, setting up the inequality $3^n \ge 900$, and testing integer values of $n$ until the inequality is satisfied. The smallest integer value of $n$ that satisfies the condition is $n=7$.

The final answer is $\boxed{7}$, which corresponds to option (B).