91Ó°ÊÓ

An inequality is a mathematical statement that shows the relationship between two expressions that are not equal. It uses symbols like <, >, ≤, or ≥ to relate expressions. In the context of our exercise, we need to determine the smallest number of digits, \( n \), necessary to form at least 900 distinct numbers using only the digits 2, 5, and 7.

The main task was to find the smallest \( n \) such that \( 3^n \), which represents the combinations, is at least 900. We express this requirement through the inequality: \[3^n \geq 900\].

This inequality sets the framework for determining the minimum number of digits needed. In this scenario, solving the inequality helps us find that the minimum \( n \) is 7, because when \( n = 7 \), \( 3^7 = 2187 \), which is more than 900, confirming that five or six digits aren't sufficient.

The concept of powers of numbers is essential for understanding how many different combinations of digits can form. A power of a number is the result of multiplying that number by itself a certain number of times. In this exercise, the digit choices (2, 5, and 7) can be independently selected for each position within an \(n\)-digit number.

This independence leads to the expression \(3^n\), which calculates the total possible combinations of \(n\)-digit numbers that can be formed. Each digit position offers 3 choices (2, 5, or 7), and if there are \(n\) positions, then there are \(3^n\) different ways to fill these positions.

• For example, if \(n = 1\), then \(3^1 = 3\): three possible single-digit numbers (2, 5, 7).
• If \(n = 2\), then \(3^2 = 9\): combinations like 22, 25, 27, etc.

Understanding powers of numbers promises efficient calculation of how many numbers can be formed with given digit choices.

A distinct number means a unique number that is different from others. The problem in our exercise involves creating 900 unique \(n\)-digit numbers using only the digits 2, 5, and 7.

When determining distinct \(n\)-digit numbers, it’s critical that no two numbers are the same. For example, 257 and 275 are considered different numbers because even though they share the same digits, their order is different, which changes the number.

As we increase \(n\), the potential for forming unique numbers increases exponentially due to the larger number of combinations. In this exercise, when we set \(n = 7\), we achieve more than 900 distinct numbers, allowing us to create a diverse set of numbers without any overlaps or repeats.

Digit combinations are the various ways in which digits can be arranged to form numbers. This exercise specifically uses the digits 2, 5, and 7 to form \(n\)-digit numbers.

Each digit position in an \(n\)-digit number represents a choice of one of the available digits. The key here is to calculate how many different \(n\)-digit numbers can be constructed by simply choosing one digit for each position. The formula \(3^n\) comes into play here, where 3 reflects the choices (2, 5, 7) per digit, and \(n\) refers to the number of digit positions.

• For example, with 2-digit numbers: We choose the first digit from 2, 5, or 7 and do likewise for the second, leading to combinations like 22, 25, and 57.
• For a 7-digit number: Every position from the first to the seventh again can independently be 2, 5, or 7.

The power of combination allows for a quick and organized way to calculate potential numbers without manually listing all possibilities.