Problem 48 - Precalculus: Mathematical for Calculus

Question

Show that \(\left(\begin{array}{l}n \\\ r\end{array}\right)=\left(\begin{array}{c}n \\ n-r\end{array}\right) \quad\) for \(0 \leq r \leq n\)

Step-by-Step Solution

Answer

The expression \( \binom{n}{r} = \binom{n}{n-r} \) is true for all \( 0 \leq r \leq n \).

Step 1: Understanding Binomial Coefficients

The binomial coefficient \( \binom{n}{r} \) is a way to calculate the number of combinations of \( n \) items taken \( r \) at a time. It is defined as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n! \) is the factorial of \( n \).

Step 2: Expressing \( \binom{n}{r} \) and \( \binom{n}{n-r} \)

The coefficient \( \binom{n}{r} \) is expressed as \( \frac{n!}{r!(n-r)!} \). Similarly, \( \binom{n}{n-r} \) is expressed as \( \frac{n!}{(n-r)!r!} \). Note that the expressions for \( \binom{n}{r} \) and \( \binom{n}{n-r} \) are identical: both are \( \frac{n!}{r!(n-r)!} \).

Step 3: Verifying Equal Expressions

Since \( r! \times (n-r)! \) is equal to \( (n-r)! \times r! \) (multiplication is commutative), the expressions \( \frac{n!}{r!(n-r)!} \) and \( \frac{n!}{(n-r)!r!} \) are indeed equal. Therefore, \( \binom{n}{r} = \binom{n}{n-r} \).

Step 4: Conclusion

We have shown that the expressions are equal, thereby proving that \( \binom{n}{r} = \binom{n}{n-r} \). This holds true for any integer \( r \) such that \( 0 \leq r \leq n \).

Key Concepts

Combinatorics: Counting Without Listing

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns within a set. Instead of listing all possible combinations, we use combinatorial techniques to count them efficiently. This is particularly useful when dealing with large sets where listing is impractical.

In our exercise, we explore how many ways we can choose a subset of items from a larger set, known as combinations. Binomial coefficients are a key concept here, representing the number of ways to choose `r` elements from a set of `n` elements. This is where the notation \( \binom{n}{r} \) comes in, also referred to as "n choose r."

• Combinatorial techniques save time and effort.
• Understanding combinations helps in real-life situations, like making a team or picking a group of friends for an event.
• Binomial coefficients connect closely with the structure of Pascal's Triangle.

Factorials: The Building Blocks of Combinatorics

The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It plays a crucial role in combinatorics, especially when calculating combinations like binomial coefficients.

For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials help us compute how many ways we can arrange elements, which is essential to determine combinations. They are used in the formula for the binomial coefficient: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]

• Factorials grow very quickly; \( 10! \) is already 3,628,800!
• They are a foundational part of permutations and combinations.
• Understanding factorials helps with grasping more complex mathematical concepts.

Mathematical Proof: Demonstrating Equalities

A mathematical proof is a logical argument demonstrating that a specific statement, proposition, or theorem, is universally true. Proofs are essential to ensure the validity of mathematical statements, showing that there are no exceptions or errors.

In our case, proving that \( \binom{n}{r} = \binom{n}{n-r} \) involves demonstrating they yield the same values through a logical sequence.

• Proof relies on understanding and applying definitions, like those of binomial coefficients and factorials.
• Using properties like the commutative nature of multiplication ensures both expressions are identical.
• Proofs help establish confidence in mathematical concepts, ensuring they are reliable.