Problem 19 - Discrete Mathematics with Applications

Question

Solve each equation, where \(n \geq 0\). $$C(n, 2)=28$$

Step-by-Step Solution

Answer

We are given the equation: $$ C(n, 2) = 28 $$ Using the binomial coefficient formula and simplifying, we get: $$ \frac{n(n-1)}{2} = 28 $$ Solving for n, we get a quadratic equation: $$ n^2 - n - 56 = 0 $$ Factoring the equation, we find two possible values for n: $$ n = 8 \quad \text{or} \quad n = -7 $$ Since \(n \geq 0\), the only valid solution is \(n = 8\).

Step 1: Write down the equation with the binomial coefficient formula

We are given the equation: $$ C(n, 2) = 28 $$ We can rewrite this using the binomial coefficient formula: $$ \frac{n!}{2!(n-2)!} = 28 $$

Step 2: Simplify the factorials

We can simplify the equation by expanding the factorials: $$ \frac{n(n-1)(n-2)!}{2(n-2)!} = 28 $$ Notice that the (n-2)! terms cancel out: $$ \frac{n(n-1)}{2} = 28 $$

Step 3: Solve for n

We need to solve the equation for \(n\). First, we can multiply both sides by 2: $$ n(n-1) = 56 $$ This gives us a quadratic equation: $$ n^2 - n - 56 = 0 $$ Now we can either factor the quadratic equation or use the quadratic formula to find the possible values for \(n\). The quadratic equation factors into: $$ (n-8)(n+7) = 0 $$ This gives us two possible values for \(n\): $$ n = 8 \quad \text{or} \quad n = -7 $$ However, we are given the constraint that \(n \geq 0\), which means n must be non-negative. Therefore, the only valid solution is: $$ n = 8 $$

Key Concepts

Quadratic Equation

Quadratic equations are a fundamental part of algebra and appear in numerous mathematical problems. They are polynomial equations of degree two, typically taking the form \( ax^2 + bx + c = 0 \). In our problem, after simplifying the binomial coefficient equation for \( n \), we arrived at the quadratic equation \( n^2 - n - 56 = 0 \). Quadratic equations have several methods of solving them, including:

• Factoring: Looking for two numbers that multiply to give the constant term \(c\) and add to give the coefficient of \(bx\).
• Completing the square: Rewriting the quadratic in the form of a perfect square trinomial.
• Quadratic formula: A universal method given by \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In the given exercise, factoring was utilized to break down \( n^2 - n - 56 \) into \( (n-8)(n+7) = 0 \). The solutions \( n = 8 \) and \( n = -7 \) emerge. Considering the condition \( n \geq 0 \), only \( n = 8 \) is a valid outcome.

Factorials

Factorials play a crucial role in various areas of math, particularly in permutations and combinations. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
In the given problem, factorials appear as parts of the formula for the binomial coefficient \( C(n, k) = \frac{n!}{k!(n-k)!} \). The exercise required finding \( C(n, 2) \), simplifying it to \( \frac{n!}{2!(n-2)!} \).
The simplification process involved canceling out common terms in the numerator and denominator, particularly \( (n-2)! \), resulting in a manageable expression \( \frac{n(n-1)}{2} \). Mastery of manipulating factorials enables efficient solving of many combinatorial and algebraic problems.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of elements within a set. It's widely used in fields like computer science, cryptography, and even everyday problems where arrangement and selection are involved.
The problem discussed involves finding the binomial coefficient \( C(n, 2) = 28 \). This represents choosing 2 items from a set of \( n \) items without regard to order. Binomial coefficients are denoted as \( C(n, k) \), calculated by the formula \( \frac{n!}{k!(n-k)!} \).

• This formula counts the number of possible combinations, hence the term "combinatorial coefficient."
• In our case, it simplifies to \( \frac{n(n-1)}{2} \), helping solve for \( n \) in terms of the given total combinations.

A solid understanding of combinatorics is essential for solving complex problems relating to probability, decision-making processes, and algorithm optimization.