Problem 56 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$C(4,1)$$

Step-by-Step Solution

Answer

4

Step 1: Understand the Combination Notation

The expression \( C(n, r) \) represents the number of ways to choose \( r \) elements from a set of \( n \) elements without regard to order. This is also known as a combination.

Step 2: Apply the Combination Formula

Use the formula for combinations: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \(n!\) denotes factorial of \(n\).

Step 3: Substitute the Values of n and r

In this problem, \( n = 4 \) and \( r = 1 \), so substitute these values into the formula:\[ C(4, 1) = \frac{4!}{1!(4-1)!} \]

Step 4: Simplify the Factorials

Calculate the factorials:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]\[ 1! = 1 \]\[ (4-1)! = 3! = 3 \times 2 \times 1 = 6 \]

Step 5: Perform the Division

Now, substitute the factorials back into the equation and simplify:\[ C(4, 1) = \frac{24}{1 \times 6} = \frac{24}{6} = 4 \]

Key Concepts

combinatorial mathematics

Combinatorial mathematics is a field of mathematics focused on counting, arrangement, and combination of elements within a set. It's crucial in various applications like probability, computer science, and optimization. At its core, combinatorial mathematics enables us to calculate the number of possible configurations that meet certain criteria.
For instance, if you want to know how many ways you can choose 3 books out of 5, combinatorial mathematics provides the tools. This area often uses techniques like permutations and combinations. Permutations consider order, while combinations do not. By understanding how elements can be selected and arranged, we can solve complex problems involving large datasets or intricate selection processes easily.

factorials

Factorials are a fundamental concept in combinatorics and algebra. The factorial of a non-negative integer n, denoted by \(n!\), is the product of all positive integers less than or equal to n. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are crucial in calculations involving permutations and combinations.
They help determine the number of ways elements can be arranged or selected from a set. For smaller integers, you can calculate factorials manually. For larger numbers, factorials grow swiftly and can become quite massive. By understanding and using factorials, complex problems in counting and arrangements can be broken down into manageable steps.

combination formula

The combination formula is often represented as \(C(n, r)\), and it calculates how many ways we can choose r items from a set of n items without considering the order. The formula is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Here’s a breakdown of how it works:
1. **Identify n and r**: Determine the total number of items (n) and the number of items to choose (r).
2. **Calculate Factorials**: Find the factorials of n, r, and (n-r).
3. **Substitute in the Formula**: Plug the values of these factorials into the combination formula.
4. **Simplify**: Perform the arithmetic to find the final number of combinations.
For example, for \(C(4, 1)\), substitute n = 4 and r = 1 into the formula: \[ C(4, 1) = \frac{4!}{1!(4-1)!} = \frac{24}{1 \times 6} = 4 \]
This means there are 4 ways to choose 1 item from a set of 4 items. Combinations are used widely in fields requiring complex selection and arrangement strategies.