Problem 93 - Precalculus with Limits

Question

In Exercises 91 - 94, prove the identity. \( _nC_{n - 1} = _nC_1 \)

Step-by-Step Solution

Answer

\(_nC_{n-1} = _nC_1\). The identity is true as proven by definition and properties of combinations.

Step 1: Understanding Combinations

Combinations are a way to count the total possible selections of items out of a larger set, irrespective of their order. In other words, we consider \(_nC_r = n! / ((n-r)!r!)\), where \(n!\) denotes the factorial of \(n\), the product of all positive integers up to \(n\). The formula is derived based on the fundamental counting principle and the idea that we want to remove duplicates that occur from the different orders in which the items can be arranged.

Step 2: Applying the definition to the left hand side of the identity

Plugging in \(n-1\) for \(r\) in the combination formula, the left-hand side of the equation, \(_nC_{n - 1}\), results in \(_nC_{n - 1} = n! / ((n-(n-1))!(n-1)!) = n! / (1!(n-1)!) = n!\).

Step 3: Applying the definition to the right hand side of the identity

Similarly, we apply the formula to the right-hand side of the equation, \(_nC_1\), which results in \(_nC_1 = n! / ((n-1)!1!) = n!\), proving the identity.

Key Concepts

Factorials in Combinatorics

When working with combinatorics, factorials are key components. A factorial, denoted as \(n!\), is the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are important because they help calculate permutations and combinations, crucial elements in counting principles.
Factorials can grow large quickly. This allows calculations involving many items and big numbers, often seen in real-world sciences or probability problems, more manageable.
A unique aspect of factorials is that \(0! = 1\), even though multiplying no numbers might seem unclear, this definition ensures math equations remain consistent and useful across different scenarios.

Understanding Combinations

Combinations help us count how many ways we can select items from a group without considering the order. Think of it as choosing a group of friends from a larger group for a team. We use the formula \(_nC_r = \frac{n!}{(n-r)!r!}\). Here, \(n\) is the total number of items to choose from, and \(r\) is the number of items we wish to pick.
This equation calculates all possible arrangements, but then divides by \((n-r)!\) and \(r!\) to remove duplicates. Why both? Because \((n-r)!\) accounts for the order of items not picked, and \(r!\) considers the order of items picked.
Combinations differ from permutations by focusing on the selection itself, not the sequence. Thus, combinations are useful in scenarios where order does not matter, such as lottery numbers or group selections.

Exploring the Counting Principle

The counting principle in mathematics is foundational for solving real-world problems efficiently. It involves systematically counting the outcomes of events that can occur by using methods like permutations and combinations.
When leveraging the counting principle, we often break complex problems into simpler stages. Each stage has a set number of outcomes that multiply together to find the total number of outcomes.

• For instance, if there are \(m\) ways to do one event and \(n\) ways to do another, there are \(m \times n\) total ways of doing both.
• Such logic forms the backbone of combinatorial calculations, leading us to the useful formulas for combinations and permutations.

With the counting principle, calculations remain straightforward and manageable, preventing errors yet providing thorough insight into problem scenarios.