Problem 51 - Precalculus Mathematics for Calculus

Question

Simplify using the Binomial Theorem. $$\text { Show that }\left(\begin{array}{l}n \\\1\end{array}\right)=\left(\begin{array}{c}n \\\n-1\end{array}\right)=n$$

Step-by-Step Solution

Answer

The identity \(\binom{n}{1} = \binom{n}{n-1} = n\) is verified using the binomial coefficient formula.

Step 1: Understanding the Binomial Coefficient Notation

The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order. It is given by the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 2: Apply the Binomial Coefficient Formula for \(k = 1\)

Using the formula, we calculate \(\binom{n}{1}\):\[\binom{n}{1} = \frac{n!}{1!(n-1)!} = \frac{n \cdot (n-1)!}{1 \cdot (n-1)!} = n.\] This shows that \(\binom{n}{1} = n\).

Step 3: Apply the Binomial Coefficient Formula for \(k = n-1\)

Next, calculate \(\binom{n}{n-1}\) using the binomial coefficient formula: \[\binom{n}{n-1} = \frac{n!}{(n-1)!1!} = \frac{n \cdot (n-1)!}{(n-1)! \cdot 1} = n.\] Thus, \(\binom{n}{n-1} = n\).

Step 4: Conclude the Equality

Since we have shown \(\binom{n}{1} = n\) and \(\binom{n}{n-1} = n\), the identities \(\binom{n}{1} = \binom{n}{n-1} = n\) are valid according to the binomial coefficient properties.

Key Concepts

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics, allowing us to determine how many ways we can select a specific number of elements from a larger set. When you see the notation \( \binom{n}{k} \), it means we are choosing \( k \) elements from \( n \). This is often read as "n choose k."

The formula to calculate the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

The exclamation point "!" represents a factorial, which we'll discuss shortly. This formula helps calculate how many different ways you can rearrange a subset of the items, ignoring the order of selection.

In our exercise, we specifically calculated for two cases: \( \binom{n}{1} \) and \( \binom{n}{n-1} \), both of which equaled \( n \). These are useful in various applications, especially when examining symmetrical properties—like knowing that choosing one item from \( n \) is directly comparable to leaving out just one item.

Combinatorics

Combinatorics is the delightful mathematics of counting, arrangement, and selection. It asks questions like "In how many ways can we choose elements?" or "How can we group things differently?" This field supports not only mathematical insights but extends its utility to computer science, probability, and logistics.

Some fundamental principles in combinatorics include:

• Permutations: Different ways to arrange a set of items. Here the order matters.
• Combinations: Selections from a set where order does not matter, directly related to binomial coefficients.

Understanding binomial coefficients gives us the power to solve combination problems quickly and effectively. For example, when you determine \( \binom{n}{1} \) or \( \binom{n}{n-1} \) both resulting in \( n \), you are using combinatorics to simplify complex counting situations, making this an invaluable tool for deeper learning and appreciation of structure and symmetry in mathematical problems.

Factorial

Factorials are foundational in math, providing the "heavy lifting" in calculations, particularly those involving permutations and combinations. Represented by \( n! \), a factorial is the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials grow rapidly and are instrumental in computing binomial coefficients.

Here's how it works in binomial coefficients:

• \( n! \): Total arrangements of \( n \) distinct items.
• \( k!(n-k)! \): Adjustments that ensure the order is not considered, limiting our interest to combinations.

The factorial function simplifies complexities by facilitating division among items and deductions as seen in our solution steps. Factorials are a pillar in understanding the mechanics that go behind both permutations and combinations, deeply enriching our approach to solving complex mathematical problems.