Problem 84 - Algebra and Trigonometry

Question

Show that $$ \left(\begin{array}{l} {n} \\ {r} \end{array}\right)+\left(\begin{array}{c} {n} \\ {r+1} \end{array}\right)=\left(\begin{array}{c} {n+1} \\ {r+1} \end{array}\right) $$

Step-by-Step Solution

Answer

The statement \(\left(\begin{array}{l} {n} \ {r} \end{array}\right)+\left(\begin{array}{c} {n} \ {r+1} \end{array}\right)=\left(\begin{array}{c} {n+1} \ {r+1} \end{array}\right) \) is proven to be true by applying the formula for the binomial coefficient and simplifying the equation.

Step 1: Identify and Write Down the Formula for the Binomial Coefficient

The binomial coefficient, usually read as 'n choose r', is given by the formula: \( \frac{n!}{r!(n-r)!} \) where '!' represents factorial.

Step 2: Apply Binomial Coefficient Formula on both sides

Plug values into the binomial coefficient formula and simplify.\n\(\left(\frac{n!}{r!(n-r)!}\right) + \left(\frac{n!}{(r+1)!(n-r-1)!}\right) = \left(\frac{(n+1)!}{(r+1)!(n-r)!}\right) \)

Step 3: Simplify the Equation

Write the left-hand side of the equation as a single fraction by finding a common denominator, then simplify.\n\( \frac{n!(n-r+1) + n!}{(r+1)!(n-r)!} = \frac{(n+1)!}{(r+1)!(n-r)!} \)\nThis simplifies to:\n\( \frac{n!(n+1)}{(r+1)!(n-r)!} = \frac{(n+1)!}{(r+1)!(n-r)!} \)

Step 4: Conclude the Proof

Since both sides of the equation are equal, it can be concluded that \( \left(\begin{array}{l} {n} \ {r} \end{array}\right)+\left(\begin{array}{c} {n} \ {r+1} \end{array}\right)=\left(\begin{array}{c} {n+1} \ {r+1} \end{array}\right) \) is indeed true.

Key Concepts

Combinatorics

When we delve into combinatorics, we find ourselves examining the various ways in which objects can be chosen, arranged, and combined. At its core, it’s about counting and arranging, often using tools like the binomial coefficients to make this complex task manageable. Combinatorics appears in everyday scenarios, such as figuring out how many different sandwiches you can make from a given set of ingredients, to more complex problems like the number of ways to seat guests at a party.

Understanding combinatorics is crucial because it forms the foundation for probability and statistics, as well as many algorithms in computer science. For students tackling homework, being familiar with combinatorial principles can transform a daunting number-crunching exercise into a solvable puzzle.

Factorial Notation

Factorial notation is a mathematical shorthand for multiplying a series of descending natural numbers. For instance, the factorial of 5, denoted as 5!, is calculated as 5 × 4 × 3 × 2 × 1. It’s fundamental in combinatorics, particularly when working with problems involving permutations and combinations, where the order of selection or arrangement matters.

Using factorial notation, the formula for binomial coefficients can be expressed succinctly, enabling swift and accurate calculations in combinatorial problems. Students should become comfortable with this notation, as it often simplifies the process and reduces mistakes in longer calculations.

Pascal's Triangle

Pascal's triangle is not only a marvel of numbers at play but also a valuable tool in combinatorics and algebra. It is constructed by aligning numbers in triangular form where each number is the sum of the two directly above it. Starting from the top with the number 1, the triangle expands downwards infinitely.

Relations to Binomial Coefficients

Each row of Pascal's triangle provides the coefficients for the expansion of a binomial equation of the form \( (a+b)^n \) and corresponds directly to binomial coefficients. This means that the triangle can be used as a quick reference for solving problems involving binomial expansions without the need for lengthy calculations.

Moreover, it embodies the relationship that was showcased in the exercise, wherein the sum of two adjacent binomial coefficients equals a binomial coefficient in the subsequent row.

Algebraic Proofs

Proving algebraic identities and equations often involves demonstrating that two expressions are equivalent under all circumstances. In the context of the given exercise, the algebraic proof serves to verify a property of binomial coefficients. By methodically breaking down the problem with factorial notation and confirming the equivalency of both sides, we achieve not just the solution to the problem, but also an insight into the interconnectedness of numbers through algebra.

For students, mastering algebraic proofs unlocks a deeper understanding of how mathematical equations work and interact. It's important to approach these proofs step by step, as jumping to conclusions without interim steps may lead to oversights or misinterpretations of the problem at hand.