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`

Explanation

To solve the problem of finding the smallest value of \( n \) such that 900 distinct \( n \)-digit numbers can be formed using the digits 2, 5, and 7, we can follow these steps:

### Step 1: Understand the formation of \( n \)-digit numbers
An \( n \)-digit number can be formed using the digits 2, 5, and 7. Since we are allowed to use these digits in each position of the number, we have 3 options (2, 5, or 7) for each digit.

### Step 2: Calculate the total number of \( n \)-digit combinations
The total number of distinct \( n \)-digit numbers that can be formed using these 3 digits is given by:
\[ 3^n \]
This is because for each of the \( n \) positions in the number, we can choose any of the 3 digits.

### Step 3: Set up the inequality
We need at least 900 distinct \( n \)-digit numbers, so we set up the inequality:
\[ 3^n \geq 900 \]

### Step 4: Solve for \( n \)
To find the smallest \( n \) that satisfies this inequality, we can calculate \( 3^n \) for different values of \( n \):

- For \( n = 1 \): \( 3^1 = 3 \)
- For \( n = 2 \): \( 3^2 = 9 \)
- For \( n = 3 \): \( 3^3 = 27 \)
- For \( n = 4 \): \( 3^4 = 81 \)
- For \( n = 5 \): \( 3^5 = 243 \)
- For \( n = 6 \): \( 3^6 = 729 \)
- For \( n = 7 \): \( 3^7 = 2187 \)

### Step 5: Identify the smallest \( n \)
From the calculations above, we see that:
- \( 3^6 = 729 \) (which is less than 900)
- \( 3^7 = 2187 \) (which is greater than 900)

Thus, the smallest value of \( n \) for which \( 3^n \geq 900 \) is:
\[ n = 7 \]

### Final Answer
The smallest value of \( n \) for which 900 distinct \( n \)-digit numbers can be formed is \( \boxed{7} \).

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