Question

If \(a, b, c\) are non-zero real constant and the number of terms in the expansion of \((a x+b y+c)^{n}\) is 36 , then \(n(n \in N)\) is equal to (1) 7 (2) 8 (3) 9 (4) 10

Short answer

The value of \(n\) is 7.

Step-by-step solution

Understand the Problem

The problem involves finding the value of the exponent \(n\) in a multinomial expansion \((a x + b y + c)^n\) such that the total number of terms is 36.

Determine Total Number of Terms Formula

In a multinomial expansion of \((a x + b y + c)^n\), the number of terms is given by the formula: \( \binom{n + 2}{2} \).

Set Up the Equation

According to the problem, the number of terms is 36. Therefore, set up the equation: \( \binom{n + 2}{2} = 36 \).

Solve the Equation for \(n\)

To solve \( \binom{n + 2}{2} = 36 \), recall that:\[ \binom{n + 2}{2} = \frac{(n+2)(n+1)}{2} \]So write the equation:\[ \frac{(n+2)(n+1)}{2} = 36 \]Multiply both sides by 2 to clear the fraction:\[ (n+2)(n+1) = 72 \]

Solve the Quadratic Equation

Now we need to solve the quadratic equation:\[ (n+2)(n+1) = 72 \]Expand the left side:\[ n^2 + 3n + 2 = 72 \]Subtract 72 from both sides to set the equation to zero:\[ n^2 + 3n - 70 = 0 \]

Factor the Quadratic Equation

Factorize the quadratic equation:\[ n^2 + 3n - 70 = 0 \]Find factors of -70 that add up to 3. These factors are 10 and -7:\[ (n + 10)(n - 7) = 0 \]

Solve for \(n\)

Set each factor to zero:\[ n + 10 = 0 \]\[ n - 7 = 0 \]So, \( n = -10 \) or \( n = 7 \). Since \( n \) must be a natural number, the solution is \( n = 7 \).

Key concepts

quadratic equations

Quadratic equations are a type of polynomial equation of degree two. They are written in the form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Solving quadratic equations often involves finding the roots or the values of \( x \) that satisfy the equation, making it equal to zero.
One common method to solve them is factoring, where you find two numbers that multiply to \( ac \) and add up to \( b \). In other cases, the quadratic formula: \[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \] is used.
This exercise involved solving a quadratic equation derived from a multinomial expansion, ultimately leading to the value of \( n \).

binomial coefficient

The binomial coefficient, often represented as \( \binom{n}{k} \), is a way of expressing the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of selection. It is calculated as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( n! \) is the factorial of \( n \), meaning the product of all positive integers up to \( n \). In the given problem, we used the binomial coefficient in a multinomial expansion which is a generalization where more than two terms are considered. The specific binomial coefficient used here was \( \binom{n+2}{2} \), representing the number of terms in the expansion. This coefficient simplifies the process of figuring out combinations and is crucial in many areas of mathematics and probability theory.

natural numbers

Natural numbers are a sequence of numbers starting from 1 and going upwards (1, 2, 3, 4, and so on). They are used for counting and ordering. In mathematics, they are sometimes also called counting numbers. An important property of natural numbers is that they are always positive.
In the given exercise, you encountered natural numbers when finding the value of \( n \). After solving a quadratic equation, we arrived at \( n = -10 \) and \( n = 7 \). Only non-negative and positive values apply because \( n \) is required to be a natural number, leading us to accept \( 7 \) as the solution.