Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

How many numbers can be formed from the digits \(1,2,3\), and 4 if repetitions are not allowed? (Note: 42 and 231 are examples of such numbers.)

Step-by-Step Solution

Answer

24 numbers can be formed.

Step 1: Understand the Problem

We need to find out how many unique numbers can be formed using each of the digits 1, 2, 3, and 4 once in each number. Repetition of digits is not allowed and we want to use all digits in each number.

Step 2: Determine the Number of Digits

We have four distinct digits: 1, 2, 3, and 4. We want to form numbers using all four digits. Therefore, each number will be a four-digit number.

Step 3: Calculate Permutations

The number of permutations of four distinct items (or digits) is given by the factorial of the number of items. Since we are using 4 digits, we calculate \(4!\).

Step 4: Compute the Factorial

Calculate \(4!\) (4 factorial), which is \(4 \times 3 \times 2 \times 1\). This results in 24. Thus, 24 different numbers can be formed.

Key Concepts

Factorial

The concept of factorial is fundamental in permutations and many areas of mathematics. Factorial, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It plays a crucial role in determining how many ways items can be arranged or ordered.

For instance, calculating \( 4! \) means multiplying all whole numbers from 1 to 4:

• \( 4 \times 3 \times 2 \times 1 = 24 \)

This tells us there are 24 different arrangements or permutations of four distinct items. Factorials grow very quickly as \( n \) increases; for example, \( 5! = 120 \), and \( 6! = 720 \). Each increase by one results in multiplying the current factorial by an additional number. Factorials are effectively used in various mathematical fields, including statistics and probability.

Distinct Digits

When considering numbers formed by a set of digits, the term 'distinct' is key. It means each digit in the set is different from the others. For example, in our problem, the digits 1, 2, 3, and 4 are distinct. Each digit can only be used once per number.

Distinct digits are important in permutations because they limit the number of possible combinations. Repetition would allow for more combinations, but distinctness restricts choices.

• First position: Any of the 4 digits
• Second position: One of the remaining 3 digits
• Third position: One of the remaining 2 digits
• Fourth position: The last remaining digit

This one-time use of each digit helps simplify calculations and ensures each number generated is unique. Thus, understanding which elements are distinct assists in applying the concept of permutations accurately.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arranging, and combining items. It provides the tools to tackle questions like the one in our exercise about forming numbers with certain requirements.

In our problem, we utilized permutations because we are arranging 4 distinct digits. Permutations consider the order of the items, making it different from combinations, which disregards order. Combinatorics helps solve problems about arranging objects or selecting items from a group, often resulting in calculations using factorial numbers.

• We ask how many ways 4 different digits can be arranged.
• The order of digits matters in forming specific numbers.
• Using the concept of permutations, we find \( 4! = 24 \) arrangements.

Combinatorics can become complex with larger sets and additional rules. However, for distinct items like our digits, factorials simplify the problem greatly. Mastery of combinatorial concepts is useful for probability, coding, and many logical problems across various disciplines.