Problem 30 - Algebra for College Students

Question

If no number contains repeated digits, how many numbers greater than 5000 can be formed by choosing from the digits \(1,2,3,4,5\), and \(6 ? 1560\)

Step-by-Step Solution

Answer

120 numbers can be formed.

Step 1: Determine the Number of Digits

To form numbers greater than 5000 using the digits \(1, 2, 3, 4, 5,\) and \(6\) without any repetition, the numbers must have at least four digits. This is because the greatest three-digit number that can be formed with these digits is 654, which is less than 5000.

Step 2: Identify the Leading Digit

To ensure that the numbers formed are greater than 5000, focus on the most significant digit (thousands place). The possible leading digits are \(5\) or \(6\), since using any digit smaller would result in a number less than 5000.

Step 3: Count Numbers Starting with 5

If the number starts with \(5\), the remaining digits \(1, 2, 3, 4, 6\) can be used. Choose 3 out of these 5 digits to complete the number. The number of ways to arrange these digits is \(5P3\) (permutation of 5 items taken 3 at a time). Calculate \(5P3 = 5 \times 4 \times 3 = 60\).

Step 4: Count Numbers Starting with 6

If the number starts with \(6\), the remaining digits are \(1, 2, 3, 4, 5\). Similarly, choose 3 out of these 5 digits to complete the number. Calculate the permutations as before: \(5P3 = 5 \times 4 \times 3 = 60\).

Step 5: Sum the Total Combinations

Add up the total numbers starting with 5 and those starting with 6. Thus, the total number of numbers greater than 5000 that can be formed is \(60 + 60 = 120\).

Key Concepts

Permutations

In combinatorics, permutations refer to the number of ways to arrange a set of items in a particular order. When dealing with permutations, every element in the set is arranged such that no two arrangements are identical.

• In the context of our problem, we're arranging the digits to form numbers.
• The order of the digits matters, which makes permutations essential.

For example, when dealing with a subset of 5 digits and selecting 3 to form a number, the formula for permutations is expressed as \(nPr\), where \(n\) is the total number of items to pick from, and \(r\) is the number of items to pick. The formula is: \[ nPr = \frac{n!}{(n-r)!} \] This represents the total ways to choose and arrange \(r\) items from \(n\) items. In our specific exercise, \(5P3 = 5 \times 4 \times 3 = 60\) possible permutations for each set of numbers starting with 5 or 6.

Digits

Digits are the basic numerical symbols, often limited to 0 through 9. They are the building blocks of numbers.

• In this exercise, we're only using the digits 1, 2, 3, 4, 5, and 6.
• Each number is a combination of these individual digits, and no digit is repeated.

Understanding digits is crucial because they form the foundation for all numbers. In mathematics, the position of a digit in a number determines its value, known as its 'place value'. For example, in the number 4523, 4 is in the thousands place, representing 4000.

Number Formation

Number formation combines individual digits to create a complete numerical value. In this problem, numbers are formed using selected digits without repetition, aiming for each number to be greater than 5000.

• The first step involves choosing the leading digit to ensure the number is above 5000.
• Next, other digits are selected to fill in the remaining positions.

For effective number formation: - Pick the appropriate leading digit (in this case, 5 or 6). - Use permutations to order the remaining available digits. In number formation, it's important that once a digit appears in a number, it cannot reappear, ensuring each digit in our sequence is unique.

Four-Digit Numbers

A four-digit number is any number composed of four individual digits, ranging from 1000 to 9999.

• In the exercise, numbers start with either 5 or 6 to guarantee they exceed 5000.
• The task is to form these four-digit numbers without repeating any of the digits.

These numbers form the basis of our objective to find how many unique permutations can be constructed while ensuring the number is sufficiently large. For example, numbers from 5000 to 5999 or from 6000 to 6999 can be formed by adjusting the remaining three digits among the provided six, without reusing any specific digit.