Problem 52 - Precalculus Mathematics for Calculus

Question

Simplify using the Binomial Theorem. $$\text { Show that }\left(\begin{array}{l}n \\\r\end{array}\right)=\left(\begin{array}{c}n \\\n-r\end{array}\right) \quad \text { for } 0 \leq r \leq n$$

Step-by-Step Solution

Answer

The identity holds because \( \binom{n}{r} = \frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!r!} = \binom{n}{n-r} \).

Step 1: Understanding the Binomial Coefficients

In combinatorics, the binomial coefficient \( \binom{n}{r} \) represents the number of ways to choose \( r \) elements from \( n \) elements without regard to the order. It is defined as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

Step 2: Introducing the Alternate Form

We are asked to show that \( \binom{n}{r} = \binom{n}{n-r} \). Begin by writing the expression for \( \binom{n}{n-r} \) using the definition: \( \binom{n}{n-r} = \frac{n!}{(n-r)!(r)!} \).

Step 3: Equating the Coefficients

Notice that the denominator of both \( \binom{n}{r} \) and \( \binom{n}{n-r} \) have the same factorial terms but in different orders. Thus, \( \frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!r!} \). These fractions are equal, proving that the two expressions are identical.

Key Concepts

Binomial Coefficient

The concept of a binomial coefficient often comes up in the context of combinatorics. It is represented as \( \binom{n}{r} \) and describes the number of ways to choose \( r \) items from \( n \) total items without considering the order. The formula for calculating a binomial coefficient is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
This equation indicates that we use factorials in its computation, which involves multiplying a sequence of descending positive numbers from a given number to 1.
The beauty of the binomial coefficient lies in its symmetry property, which states: \( \binom{n}{r} = \binom{n}{n-r} \). This means, the number of ways to choose \( r \) items is the same as choosing \( n-r \) items, given the total is \( n \).
The symmetry makes conceptual sense because choosing \( r \) items automatically leaves \( n-r \) items unchosen; hence the two are complementary.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing configurations. The binomial coefficient is a fundamental idea in this field.
Understanding combinatorics helps us solve problems where we need to determine the number of possible arrangements or selections of objects. Combinatorial problems can be simple, like choosing fruits from a basket, or more complex like arranging speakers at a conference.
Key concepts in combinatorics include:

• Permutations: Arrangements where order matters.
• Combinations: Selections where order does not matter.
• Binomial Theorem: Provides a formula to expand expressions raised to a power.

In many scenarios, we rely on combinatorial principles, like the binomial theorem, to simplify complex counting problems.
The binomial theorem, for example, leverages the binomial coefficients in its expansion formula, making it a potent tool in tackling polynomial expressions.

Factorials

Factorials are mathematical expressions represented by an exclamation mark, such as \( n! \), which denotes the product of all positive integers up to \( n \). This may look like: \( n! = n \times (n-1) \times ... \times 2 \times 1 \).
Factorials have a crucial role in determining binomial coefficients. They help us calculate the number of ways to arrange or select objects by scaling up rapidly with larger numbers.
A few important properties of factorials include:

• Zero Factorial: By convention, \( 0! = 1 \). This simplification helps in calculations involving combinations when \( r \) or \( n-r \) equals zero.
• Recursive relation: Factorials are constructed recursively: \( n! = n \times (n-1)! \).

Factorials, when used in the binomial coefficient formula, allow us to deeply understand the calculation of combinations and permutations, as seen in both basic and complex combinatorial challenges.