Question

How many positive integers less than 1,000,000 can be written using only the digits 7 , 8 , and 9 ? How many using only the digits \(0,8,\) and \(9 ?\)

Short answer

1092 numbers using 7, 8, 9; 728 numbers using 0, 8, 9.

Step-by-step solution

Understanding the Problem for Digits 7, 8, and 9

We need to count numbers made only with the digits 7, 8, and 9 that are less than 1,000,000. The number of digits each number can have ranges from 1 to 6 since 1,000,000 has 7 digits, which allows for numbers up to 999,999.

Counting for Each Digit Length (using 7, 8, 9)

For each possible digit count (1 to 6), calculate the number of numbers that can be formed. For any digit place, there are 3 choices (either 7, 8, or 9). Hence, for a 1-digit number, the possibilities are 3; for a 2-digit number, the possibilities are 3^2, and so on up to 3^6 for a 6-digit number.

Calculating Total Possibilities (using 7, 8, and 9)

Sum the number of possibilities for each digit length: \(3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 = 3 + 9 + 27 + 81 + 243 + 729 = 1092.\)

Understanding the Problem for Digits 0, 8, and 9

Similarly, we count numbers made only with digits 0, 8, and 9, which are less than 1,000,000. Digit counts still range from 1 to 6, but the use of '0' means we cannot use it in the highest place value position.

Counting for Each Digit Length (without leading 0)

For numbers without leading zeros, calculate possibilities for each digit length. For digits in the highest place, only 8 or 9 can be used. For a 1-digit number, 2 possibilities; for a 2-digit number, 2 choices for first digit and 3 for the second, i.e., 2 * 3^1 possibilities, continuing until 2 * 3^5 for 6-digit numbers.

Calculating Total Possibilities (using 0, 8, and 9)

Sum up all calculated possibilities: \(2^1 + 2 imes 3^1 + 2 imes 3^2 + 2 imes 3^3 + 2 imes 3^4 + 2 imes 3^5 = 2 + 6 + 18 + 54 + 162 + 486 = 728.\)

Key concepts

Permutations

Permutations are a key concept in combinatorics, used to count the number of ways to arrange a set of items. When discussing permutations, the order of these items matters.
For example, considering the digits 7, 8, and 9, each unique sequence of these digits forms a different permutation.
In the context of the exercise, when forming numbers up to 999,999, each potential slot in our number can be filled with any of the available digits.

• A 1-digit number formed with these digits results in 3 permutations, namely 7, 8, or 9.
• A 2-digit number allows for each position to have a choice among the three digits, giving us a total of \(3^2 = 9\) permutations.
• Similarly, increasing the digit length continues this multiplying pattern up to \(3^6 = 729\) for a 6-digit sequence.

Understanding how permutations work enables us to break down complex counting problems into simpler parts.

Counting Principles

Counting principles help solve combinatorial problems by simplifying the way we calculate possible outcomes. A basic rule of thumb is the multiplication principle, which states that if you have multiple stages of selections, you multiply the number of choices at each stage.
For instance, when calculating the total number of numbers that can be formed from the digits 7, 8, and 9:

• We explore multiple subsets of the problem, such as 1-digit, 2-digit numbers, etc., up to 6-digit numbers.
• In each case, counting is done by raising the number of digit choices (3 choices: 7, 8, 9) to the power of the number of positions available, which results in \(3^n\) choices for any number of digits.
• The sum of these choices gives us the total count of valid numbers up to 999,999.

By understanding counting principles, we can solve these problems quickly, whether it involves selecting items, forming permutations, or other combinatorial scenarios.

Mathematical Problem Solving

Mathematical problem solving involves analyzing a problem, breaking it down into manageable parts, and systematically addressing each one.
In solving the exercise, the analysis begins by understanding the constraints and goals — numbers built from specific digits and no more than 6 digits total.

• First, identify what the problem asks: form numbers using the digits and ensure they are below a certain value.
• Then, assess the possible choices for each numerical position, depending on whether leading zeros are allowed, like with digits 0, 8, and 9.
• This methodical approach ensures that each option is accounted for without overcounting, leading to a correct total.

Problem-solving in mathematics often requires creativity for approaching new scenarios, combined with logical reasoning to compute valid solutions. Through practice, this skill becomes highly proficient, leading to effective solving of even complex combinatorial problems.