Problem 29 - Algebra and Trigonometry

Question

List all the combinations of 4 objects \(1,2,3,\) and 4 taken 3 at a time. What is \(C(4,3) ?\)

Step-by-Step Solution

Answer

There are 4 combinations: \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}.

Step 1: Understand the Combination Formula

The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] This represents the number of ways to choose \(r\) objects from a set of \(n\) objects without regard to the order of selection.

Step 2: Substitute Values into the Formula

For this problem, \(n = 4\) and \(r = 3\). Substitute these values into the combination formula: \[ C(4, 3) = \frac{4!}{3!(4-3)!} \]

Step 3: Simplify the Factorials

Calculate the factorials: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] \[ 3! = 3 \times 2 \times 1 = 6 \] \[ 1! = 1 \] So, the formula becomes: \[ C(4, 3) = \frac{24}{6 \times 1} = \frac{24}{6} = 4 \]

Step 4: List the Combinations

The different ways to choose 3 objects out of 4 (1, 2, 3, 4) are: \[ \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\} \]

Key Concepts

Combinatorial Mathematics

Combinatorial mathematics is a fascinating branch of mathematics focused on counting and arranging objects. It deals with combinations, permutations, and other ways to organize a set of items. Here, you learn how to choose items from a set in specific ways. For instance, combinations are about selecting items where the order does not matter, unlike permutations where order does matter. Understanding this distinction is crucial for solving many problems in different fields like statistics, computer science, and even biology.

By applying combinatorial mathematics, you can solve problems related to grouping, scheduling, and resource allocation efficiently.

Factorials

Factorials are fundamental in many combinatorial calculations. A factorial of a number (denoted as \( n! \) ) is the product of all positive integers up to that number. For example:\[ 4! = 4 \times 3 \times 2 \times 1 = 24. \3! = 3 \times 2 \times 1 = 6. \1! = 1 \ \]The concept of factorials finds its applications in various mathematical contexts, especially in combinations and permutations.

When computing combinations, factorials help break down and simplify the calculations. Although the numbers and their multiplication might seem intimidating, they are essential in providing accurate results in combinatorial mathematics.

Combination Formula

One of the key ideas in combinatorics is the combination formula. The formula to calculate combinations is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items and \( r \) is the number of items to choose.

This formula gives you the number of ways to choose \( r \) items from a set of \( n \) items without regard to order. For example, if you need to find how many ways you can choose 3 objects from a set of 4 (1, 2, 3, 4), you use the combination formula: \[ C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{24}{6 \times 1} = 4 \] This calculation reveals that there are 4 ways to choose 3 objects out of 4.

The formula is very powerful and can be adapted to different problems and scenarios, making it a vital tool in solving complex combinatorial questions.