Problem 122 - A Complete Resource Book in Mathematics for JEE Main

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) 5 (B) 6 (C) 7 (D) 8

Step-by-Step Solution

Answer

The smallest value of \( n \) for which 900 distinct numbers can be formed is 7.

Step 1: Understanding the Problem

We need to find the smallest number of digits \( n \) such that 900 distinct numbers can be formed using only the digits 2, 5, and 7. Each digit must be used and numbers cannot repeat.

Step 2: Determine Possible Numbers Per Digit Count

For a number with \( n \) digits, and each digit having 3 possible choices (2, 5, or 7), the total number of distinct \( n \)-digit numbers is \( 3^n \).

Step 3: Calculate Minimum \( n \) for 900 Numbers

We need to find the smallest \( n \) such that \( 3^n \geq 900 \). This means solving for \( n \) in the inequality \( 3^n \geq 900 \).

Step 4: Check with Small Values of \( n \)

Calculate values of \( 3^n \) for small \( n \):- \( 3^5 = 243 \)- \( 3^6 = 729 \)- \( 3^7 = 2187 \)Since \( 3^6 = 729 \) is less than 900, and \( 3^7 = 2187 \) is greater than 900, the smallest \( n \) can be determined.

Step 5: Verify the Solution

With \( 3^7 = 2187 \) greater than 900, but \( 3^6 < 900 \), it's confirmed that \( n = 7 \) is the minimum number of digits needed.

Key Concepts

Understanding n-digit numbers

An \( n \)-digit number is a positive number that consists of exactly \( n \) digits. This is a simple but important concept in mathematics and number theory. When you think of an \( n \)-digit number, you should imagine a number that takes up \( n \) places, with each place taken by a digit from 0 to 9. Examples of an \( n \)-digit number include:

• A 3-digit number like 256
• A 5-digit number like 12345
• An \( n \)-digit number where \( n = 7 \) might be 7654321

These numbers are important because different \( n \)-digit numbers may have different properties, which can be used in mathematical problems such as determining distinct numbers, as discussed in our problem.

Exploring distinct numbers

Distinct numbers refer to numbers that are unique from each other. When forming distinct \( n \)-digit numbers, it's essential to ensure that no two numbers are identical. For instance, in our exercise, we're using only the digits 2, 5, and 7 to form distinct numbers. Here are some key points about distinct numbers:

• The number of distinct \( n \)-digit numbers depends on the available choices for each digit.
• For a problem requiring distinctness with only certain digits, each number must be checked to ensure it hasn't been used before.
• Understanding how many possible distinct numbers can arise is crucial for solving probs