Problem 5 - Calculus One Again

Question

Prove that $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n}{k}\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right) $$

Step-by-Step Solution

Answer

The identity is proven as both sides simplify to \( \binom{n}{k} \).

Step 1: State the Binomial Coefficient Formula

The binomial coefficient \( \binom{n}{k} \) is defined as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This represents the number of ways to choose \( k \) elements from a set of \( n \) elements.

Step 2: Simplify the Right-Hand Side Expression

Consider the right-hand side of the equation, \( \frac{n}{k} \binom{n-1}{k-1} \). By the definition of binomial coefficient, \( \binom{n-1}{k-1} = \frac{(n-1)!}{(k-1)!(n-k)!} \). Therefore, \( \frac{n}{k} \binom{n-1}{k-1} = \frac{n}{k} \times \frac{(n-1)!}{(k-1)!(n-k)!} \).

Step 3: Multiply the Expression

Multiply \( \frac{n}{k} \) with \( \frac{(n-1)!}{(k-1)!(n-k)!} \). This results in \( \frac{n \cdot (n-1)!}{k \, (k-1)! \, (n-k)!} \).

Step 4: Simplify the Expression

Since \( n \cdot (n-1)! = n! \), this results in \( \frac{n!}{k!(n-k)!} \), which is the definition of the binomial coefficient \( \binom{n}{k} \).

Step 5: Conclusion

Having simplified \( \frac{n}{k} \binom{n-1}{k-1} \) to \( \binom{n}{k} \), the given identity is proven.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental concept in algebra that describes the expansion of powers of a binomial. A binomial is a mathematical expression that has two terms, such as \((x + y)^n\). The theorem gives us a way to expand this expression into a sum of terms of the form \(a \cdot x^{b} \cdot y^{c}\), where \(a\), \(b\), and \(c\) are non-negative integers. The coefficients of these terms are given by the binomial coefficients \(\binom{n}{k}\).

• Each coefficient represents the number of ways to choose \(k\) elements from a set of \(n\) elements.
• This is precisely the definition used in the binomial theorem: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} \cdot x^{n-k} \cdot y^k\).

The power and utility of the binomial theorem lie in its ability to simplify polynomial expressions and solve problems in various fields of mathematics.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns. It plays a crucial role in probability theory, statistics, and computer science. At the heart of combinatorics lies the binomial coefficient, which is used to determine how many different combinations can be made from a set. When choosing \(k\) elements from a set of \(n\), the number of possible combinations is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) represents the factorial, or the product of all positive integers up to \(n\).

• This principle helps solve problems related to permutations, combinations, and partitions.
• Combinatorics is used in game theory, coding theory, and optimal decision-making algorithms.

Understanding combinatorics and binomial coefficients makes it easier to approach problems involving probability and organized counting.

Mathematical Proof

A mathematical proof is a logical argument demonstrating the truth of a mathematical statement. It is an essential part of mathematics because it ensures that the results we obtain are correct and reliable. Proving something mathematically often involves using already established facts, definitions, and logical reasoning. In the case of proving the identity \(\binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1}\), we use principles from both algebra and combinatorics.

• The proof starts by using the definition of the binomial coefficient and expressing both sides using factorial notation.
• Simplifying the expressions step by step and verifying that they match leads to a successful proof.

By understanding mathematical proofs, we can build confidence in our mathematical reasoning and tackle more complex problems with clarity.