Question

The following are the steps involved in solving \({ }^{\mathrm{n}} \mathrm{c}_{2}=36\) for \(\mathrm{n}\). Arrange then in sequential order. (a) \(\mathrm{n}^{2}-\mathrm{n}-72=0\) (b) As \(\mathrm{n}>0, \mathrm{n}=9\) (c) \(\mathrm{n}=9, \mathrm{n}=-8\) (d) \((n-9)(n+8)=0\) (e) \(\frac{\mathrm{n}(\mathrm{n}-1)}{1 \times 2}=36\) (1) eacbd (2) eadcb (3) eadbc (4) edcdb

Short answer

As a teacher, create a short answer based on the following step-by-step solution: Question: Given the equation ${ }^nc_2=36$, solve for n. Answer: When solving for n in the equation ${ }^nc_2=36$, we first rewrite the equation using the combination formula, which results in $\frac{n(n-1)}{2} = 36$. We then simplify the equation as $n(n-1) = 72$. Next, we expand the expression and set the equation to zero: $n^2 - n - 72 = 0$. We can factor this quadratic equation as $(n-9)(n+8)=0$. Solving for n, we get two solutions: $n=9$ and $n=-8$. Since n must be a positive integer in this context, we choose $n=9$ as the solution.

Step-by-step solution

Rewrite the equation using the combination formula

We know that \({ }^nc_2\) represents the number of combinations, which can be written as \(\frac{n(n-1)}{2}\). So, we can rewrite the given equation as: \(\frac{n(n-1)}{2} = 36\).

Simplify the equation

To get rid of the fraction, we can multiply both sides of the equation by 2, resulting in: \(n(n-1) = 72\).

Expand the expression and move all terms to one side

Expand the left side of the equation: \(n^2 - n = 72\). Now, move all terms to one side of the equation, which results in a quadratic equation: \(n^2 - n - 72 = 0\).

Factor the quadratic equation

We can factor the quadratic equation \((n^2 - n - 72)\) as \((n - 9)(n + 8) = 0\).

Solve for n

Having factored the quadratic equation, we can solve for n by setting each factor to zero: \(n - 9 = 0\) and \(n + 8 = 0\). This gives us two solutions: \(n = 9\) and \(n = -8\).

Consider the context of the problem

Since n represents the number of elements in a set, it must be a positive integer. Therefore, we can eliminate the negative solution and conclude that \(n = 9\). The correct order of the steps is (3) eadbc.

Key concepts

Combination Formula

In mathematics, the combination formula is a fundamental concept often used in probability and combinatorics to determine the number of ways a certain number of items can be selected from a larger group without regard to the order. This is formally presented as c where n represents the total number of items and r represents the number of items being chosen. The factorial notation, denoted by an exclamation mark (!), indicates the product of all positive integers up to that number.When we look at an expression like , it represents the number of different combinations of n items taken 2 at a time. To solve for n when the number of combinations is known, as in the textbook exercise, we can express this formula as: c This is a practical application of the combination formula, using algebra to find a variable within the context of combinatorial analysis. Understanding this concept is vital for students tackling problems related to probability, statistics, and other fields that involve counting and arrangement.

Solving Quadratic Equations

Quadratic equations form an integral part of algebra and are characterized by the highest degree of the variable being two. The standard form of a quadratic equation is To solve these equations, one can apply various methods like graphing, using the quadratic formula, completing the square, or factorization. The method illustrated in the original exercise involves transforming the combination equation into a quadratic one and then factoring it, which is a straightforward and effective technique when the equation is factorable.Although not every quadratic equation can be factored easily, recognizing factorable patterns is an essential skill that simplifies problem-solving. The key to factorization lies in finding two numbers that multiply to give the constant term (c) and add up to the coefficient of the x term (b). When successful, this method leads to a quick and elegant solution.

Factorization

Factorization is the process of breaking down an algebraic expression into the simplest building blocks, or 'factors', that, when multiplied together, give the original expression. It is a crucial skill in algebra, especially when solving quadratic equations. Whenever we have an expression such as ,the goal is to find two binomials whose product results in the original quadratic equation.In our textbook exercise, the expression is factored into . This process essentially reverses multiplication and is often one of the first techniques attempted when solving quadratics. Successful factorization depends on recognizing patterns and applying algebraic identities. These identities include the difference of squares, the perfect square trinomials, and others. Mastering factorization can make a vast array of mathematical problems more manageable.

Positive Integer Solutions

In mathematics, especially in combinatorics and number theory, finding positive integer solutions is a common requirement. A positive integer is any whole number greater than zero. When confronted with solutions to a problem that include both positive and negative integers or non-integer numbers, context is key to determining which are viable. In a problem involving combinations, such as calculating ,we inherently deal with countable items. Since you can't have a negative number of items, we discard negative solutions. As noted in the textbook exercise, after applying the combination formula and solving the quadratic equation, we obtain two values for n. However, given that we are considering combinations where n must represent the number of items (a positive integer), the negative solution ( ) is rejected, leaving us with the only contextually appropriate answer ( ). This logical reasoning is vital when filtering through potential solutions and finding those that make sense within the confinements of the problem presented. Understanding how to identify and restrict solutions to positive integers is essential for solving a wide variety of mathematical problems.