Problem 17 - Algebra and Trigonometry

Question

13–18 Evaluate the expression. $$C(100,1)$$

Step-by-Step Solution

Answer

100

Step 1: Understand the Expression

The expression given is \( C(100, 1) \), which is a combination. The combination formula is used to find the number of ways to choose \( k \) items from \( n \) total items without regard to order. It is written as \( C(n, k) \).

Step 2: Apply the Combination Formula

The formula for combinations is \( C(n, k) = \frac{n!}{k!(n-k)!} \). Here, \( n = 100 \) and \( k = 1 \). Substitute these values into the formula: \( C(100, 1) = \frac{100!}{1!(100-1)!} \).

Step 3: Simplify the Factorials

We know that \( 1! = 1 \) and \( (100-1)! = 99! \). Therefore, the expression simplifies to \( \frac{100!}{1 \times 99!} \). Since \( 100! = 100 \times 99! \), substitute and simplify: \( \frac{100 \times 99!}{99!} = 100 \).

Step 4: Final Result

The simplification shows that \( C(100, 1) = 100 \). This means there are 100 different ways to choose 1 item from 100 items.

Key Concepts

Factorials

Factorials are an essential part of combinatorics and mathematical expressions. They are represented by an exclamation mark (!) following a number and signify the product of all positive integers up to that number. For example, the factorial of 4, written as 4!, is equal to 4 x 3 x 2 x 1, which equals 24.
Factorials are vital for calculating combinations and permutations, as they help determine the number of possible arrangements or selections.

• For any integer \( n \), \( n! \) is calculated as \( n \times (n-1) \times (n-2) \times \,\ldots\, \times 2 \times 1 \).
• Special cases include \( 0! \), which is defined to be 1.
• Factorials grow extremely fast as \( n \) increases, which is why computations involving them often entail simplification.

Factorials play a crucial role in simplifying mathematical expressions, like those encountered in combinatorics. Understanding this concept aids in grasping the logic behind combination formulas.

Combinatorics

Combinatorics deals with counting, arranging, and finding patterns within sets. When solving problems like \( C(100, 1) \), combinatorics provides the tools for understanding how many ways we can choose or arrange items.
The most common aspects of combinatorics include permutations and combinations. Permutations consider the order of arrangement, while combinations do not:

• Permutations: Used when the order of selection matters. Calculated with \( nPr = \frac{n!}{(n-r)!} \).
• Combinations: Used when the order of selection does not matter. Calculated with \( C(n, k) = \frac{n!}{k!(n-k)!} \).

In our exercise, \( C(100, 1) \) signifies choosing 1 item from a set of 100, showcasing how combinatorics helps solve real-world selection problems without considering the arrangement's sequence.

Mathematical Expressions

Mathematical expressions are combinations of numbers and symbols that represent a value or process. In solving problems like \( C(100, 1) \), understanding mathematical expressions is pivotal as it allows us to correctly interpret and manipulate these symbols to find a solution.
Expressions can include operators (such as addition, multiplication), numbers, and parentheses to indicate the order of operations. In combinatorics, mathematical expressions are used heavily to formalize the counting process:

• Formulas, like \( C(n, k) = \frac{n!}{k!(n-k)!} \), are expressions providing a clear method to calculate specific mathematical tasks.
• Simplification of expressions, such as reducing \( \frac{100!}{99!} \) to 100, is essential for finding practical solutions.
• Expressions help to generalize the problem-solving process, making them adaptable to various scenarios and sizes of data sets.

Understanding mathematical expressions in the context of combinatorics and beyond enables students to break down complex problems into simpler, solvable parts, leading to accurate and efficient solutions.