Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Step-by-Step Solution

Answer

The smallest value of \( n \) is 7.

Step 1: Understand the Problem

We need to determine the smallest number of digits, which we will call \( n \), that a number can have such that 900 distinct numbers can be formed using the digits 2, 5, and 7.

Step 2: Calculate the Number of Possible Combinations

Each digit in the \( n \)-digit number can be one of 2, 5, or 7. Hence, for an \( n \)-digit number, there are \( 3^n \) possible combinations that can be formed.

Step 3: Set Up the Inequality

To find the smallest \( n \) such that at least 900 distinct numbers can be formed, solve \( 3^n \geq 900 \).

Step 4: Evaluate the Condition for Different \( n \) Values

- For \( n = 6 \), \( 3^6 = 729 \), which is less than 900.- For \( n = 7 \), \( 3^7 = 2187 \), which is greater than 900.Thus, \( n = 7 \) is the smallest integer that satisfies \( 3^n \geq 900 \).

Step 5: Conclusion

The smallest value of \( n \) for which 900 distinct \( n \)-digit numbers can be formed using only the digits 2, 5, and 7 is 7.

Key Concepts

Distinct Numbers

In the world of combinatorics, a "distinct number" refers to a number that is unique within a given set. The exercise at hand deals with forming n-digit numbers, where each entire number must be different from the others.
To achieve this, we focus on combinations of the digits 2, 5, and 7. Whenever you meet a problem involving distinct numbers, think about:

• Ensuring no repetitions across the whole set of numbers.
• Considering every digit's position has importance, affecting the overall identity.

The challenge stems from ensuring distinctiveness while following restrictions, such as limited digit options and a requirement for a specific number, like 900 in this case. This application of combinatorial principles ensures each number remains unique, thereby fostering a comprehensive understanding of distinct numbers used in various mathematical contexts.

Digit Combinations

Digit combinations refer to the various ways digits can be arranged to form numbers. In this problem, we are interested in how many different numbers we can form using the available digits: 2, 5, and 7. Each digit combination results in a different number.
Here’s what you should keep in mind about digit combinations:

• Each position in an n-digit number can independently be any of the given digits.
• The number of total combinations is calculated as the base number (3 in this scenario for the 3 digits) raised to the power of the number of digits (n).

In practical terms, if you want to form an n-digit number where all digits come from choices of 2, 5, and 7, you will find all possible numbers by calculating \(3^n\). This helps anticipate just how many unique numbers can be produced. As demonstrated, if you want at least 900 numbers, you need to explore for an n that achieves this.

Inequality Problem Solving

Inequality problem solving is about determining the range or threshold requirements that your solution must satisfy. For the case of forming distinct digit combinations, our inequality ensures that the number of distinct combinations is equal to or greater than a required figure—in this instance, 900.
Here's a simplified approach:

• Set up an inequality that matches the requirements: like \(3^n \geq 900\).
• Test integer values of n to find the smallest one that meets the inequality (e.g., calculate \(3^6 = 729\) and \(3^7 = 2187\)).
• Identify the smallest n that fits, ensuring the total possible combinations meet or exceed 900.

Using inequalities effectively helps you pinpoint the smallest solution that satisfies your conditions without manually checking every possible scenario. This is particularly useful when dealing with larger numbers or complex conditions, such as potentially infinite or exceedingly vast combinations.