Problem 61 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(C(5,0)\)

Step-by-Step Solution

Answer

The value of \(C(5,0)\) is 1.

Step 1: Understand the Notation

The notation \(C(n,k)\) stands for the binomial coefficient, which is the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order. This can also be called 'n choose k.'

Step 2: Apply the Formula

The formula for the binomial coefficient is \(C(n,k) = \frac{n!}{k!(n-k)!}\). For \(C(5,0)\), substitute \(n = 5\) and \(k = 0\) into the formula.

Step 3: Simplify the Factorials

Calculate the factorials: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) and \(0! = 1\) because the factorial of zero is defined as 1; \((5-0)! = 5! = 120\).

Step 4: Compute the Division

Substitute into the formula to get \(C(5,0) = \frac{5!}{0!(5-0)!} = \frac{120}{1 \times 120} = 1\).

Key Concepts

Factorial

The factorial is a mathematical operation that is crucial in many areas, including binomial coefficients. It is represented by the symbol "!" following an integer. To understand how it works, follow this simple rule: the factorial of a number, say "n", is the product of all positive integers less than or equal to "n". For example, the factorial of 5 is written as 5! and calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]Interestingly, there is a unique case with the factorial of zero. By definition, 0! is equal to 1. This may seem unusual at first, but it is defined this way to make many mathematical formulas work consistently, particularly when dealing with combinations and permutations. Understanding factorials is essential when solving problems related to permutations, combinations, and the binomial theorem.

Combinatorics

Combinatorics is the area of mathematics that studies counting, arranging, and finding patterns. It's especially useful in fields like probability, statistics, and computer science. One of the key aspects of combinatorics is that it helps us count things in an efficient way without having to explicitly enumerate all possibilities. Some important concepts within combinatorics include:

• Permutations: Arrangements where order matters.
• Combinations: Selections where order does not matter.
• Partitions: Ways of breaking a set into subsets.

At the heart of combinatorics lies the binomial coefficient, which we often express as "n choose k". This helps us determine how many ways we can select k items from n items without considering order. This is exactly what the original exercise and solution are illustrating.

n choose k

The notation "n choose k", written as \(C(n, k)\) or sometimes as \(\binom{n}{k}\), refers to the binomial coefficient. It represents the number of ways to choose k items from a set of n items without paying attention to the order of selection. This is calculated using the formula:\[ C(n,k) = \frac{n!}{k!(n-k)!} \]For instance, in the exercise \(C(5,0)\), it means choosing 0 items from a set of 5. Applying the formula:\[ C(5,0) = \frac{5!}{0!(5-0)!} = \frac{120}{1 \times 120} = 1 \]This result makes intuitive sense: there is exactly one way to choose nothing from a group of items, which illustrates a fundamental aspect of combinations. The simplicity of this result underscores the elegance of combinatorics and the utility of binomial coefficients in solving problems involving selections and arrangements.