If 4 th term in expansion of \([p x+(1 / x)]^{n}\) is constant then \(n=\) (a) 3 (b) 4 (c) 5 (d) 6

One of the key elements of the binomial theorem is the binomial coefficient, which is represented as \( {n \choose k} \) or more commonly \( \binom{n}{k} \). This notation stands for the number of ways to choose \( k \) elements out of a set of \( n \) distinct elements without considering the order of selection. Mathematically, it is expressed as \( \frac{n!}{k!(n-k)!} \), where \( n! \) denotes the factorial of \( n \) and illustrates the product of all positive integers up to \( n \).

The significance of binomial coefficients emerges in the binomial expansion, where they determine the weight of each term. As an exercise improvement advice, understanding and calculating binomial coefficients are crucial for expanding a binomial expression correctly and is the first step to identifying individual terms in the binomial expansion.

In polynomials, a constant term is the term with the highest power of zero, i.e., devoid of any variables. This is the term that remains unchanged despite the value of the variable. When dealing with a binomial expansion, finding the constant term involves setting the variable to a power of zero, effectively making it 'disappear' from the term.

In the context of the given exercise, detecting the constant term helps to determine the particular power to which the binomial must be raised. When students come across phrases like 'the nth term is constant,' it is a cue to equate the variable's power to zero and solve for \( n \). This method isolates \( n \) and simplifies the process of finding the correct term without working through the complete binomial expansion.

The binomial theorem unfolds the powerful pattern hidden within binomials raised to a power. It states that the expansion of \((a + b)^n\) is composed of terms that follow a specific order, each with a corresponding binomial coefficient, and powers of \(a\) and \(b\) that sum to \(n\).

Understanding how to identify individual terms in the expansion is essential. To find the \(k^{th}\) term, or \(T_{k+1}\), one can use the formula \( \binom{n}{k} a^{n-k} b^k \), by plugging in \(n\) (the power to which the binomial is raised), \(a\), \(b\), and \(k\) (which is one less than the term number since counting starts from zero). By doing so, one can systematically determine any term in the binomial expansion without fully expanding the expression, saving time especially on tests or exams.