It coefficients of middle terms in expansion of \((1+\lambda \mathrm{x})^{8}\) and \((1-\lambda \mathrm{x})^{6}\) are equal then \(\lambda=\) (a) \((2 / 7)\) (b) \([(-2) / 7]\) (c) \([(-3) / 7]\) (d) None of these

The Binomial Theorem is a powerful tool in algebra that allows you to expand expressions raised to a power in the form \( (a + b)^n \). In general, the theorem states that this expression can be expanded into the sum of terms in the form \( {n\choose k} a^{n-k} b^k \), where \( k \) ranges from 0 to \( n \) and \( {n\choose k} \) represents the binomial coefficient. Determining these coefficients and understanding the expanded form can significantly simplify calculations and the solving of algebraic problems.

For instance, when expanding \( (1 + \lambda x)^8 \), the binomial theorem tells us that we get a series of terms from \( k = 0 \) to \( k = 8 \) where each term has coefficients determined by the binomial coefficients. This methodology is applied to identify and equate middle terms in problems involving binomial expansions to find specific unknown values like \( \lambda \) in our exercise.

The binomial coefficient, denoted as \( {n\choose k} \), is a fundamental concept in the realm of combinatorics. It represents the number of ways to choose \( k \) elements from a set of \( n \) distinct elements without considering the order of selection. This coefficient can be calculated using the formula:
\[ {n\choose k} = \frac{n!}{k!(n-k)!} \] where \( ! \) denotes the factorial operation. In our context, to find the coefficients of the middle terms of the binomial expansions, we apply this formula to compute values like \( {8\choose 4} \) and \( {6\choose 3} \) which are crucial to solving the problem at hand.

The significance of the binomial coefficient extends beyond algebra to various fields such as probability and statistics, where it is used to calculate combinations and probabilistic events.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables like \( x \) and \( y \) and the operations of addition, subtraction, multiplication, division, exponentiation, and root extraction. Unlike an equation, an algebraic expression doesn't have an equality sign; it is not a statement of equality but rather a component that can be part of a broader equation or function.

In the exercise where we are equating middle terms, \( (1+\lambda x)^8 \) and \( (1-\lambda x)^6 \) are algebraic expressions that are being expanded using the binomial theorem. The careful manipulation of these expressions – which involves recognizing the similar and dissimilar terms – is vital to solving algebraic problems, especially when we are concerned with finding the value of a variable, in this case \( \lambda \).

Combinatorics is a branch of mathematics dealing with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, among others. Within combinatorics, one often encounters problems related to the counting of combinations, which is directly connected to the binomial coefficient we discussed earlier.

When we encounter the task of equalizing the coefficients of the middle terms of binomial expansions, combinatoric principles come into play to determine the number of combinations possible, and thereby the coefficients themselves. Understanding these underlying combinatoric concepts can vastly enhance a student's ability to tackle a wide array of problems involving binomial expressions and their applications.