If rth term in the expansion of \(\left[a x^{p}+\left(b / x^{q}\right)\right]^{n}\) is constant, then \(\mathrm{r}=\) (a) \([(q n) \bar{m}+q)]+1\) (b) \([(\mathrm{pn}) /(\mathrm{p}-\mathrm{q})]+1\) (c) \([(\mathrm{pn}) /(\mathrm{p}+\mathrm{q})]+1\) (d) \([(\mathrm{pn}) /(\mathrm{p}-\mathrm{q})]-1\)

When it comes to the Binomial Expansion, it's essential to understand that it's a way of expressing the powers of a binomial as a sum of terms. It follows a specific pattern where each term has a coefficient, two types of variables (if there are any), and exponents that align with the position of the term in the expansion.

The binomial theorem states that \( (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \) where \( {n \choose k} \) are the binomial coefficients, and the exponent on \( a \) decreases while the exponent on \( b \) increases with each term. This is crucial because in any binomial expansion, when you wish to find a specific term, like the rth term in the exercise, knowing how to write it out using the theorem helps simplify and solve problems effectively.

In algebra, Combining Like Terms is a fundamental technique used to simplify expressions or equations. Like terms are terms that have exactly the same variable parts, which means their variables and their exponents are the same. While combining, you simply add or subtract the coefficients of these terms.

For example, in the binomial expansion exercise, after applying the binomial theorem, you'll find different powers of \( x \) that need to be combined. By adding together the exponents of \( x^{p(n-r)} \) and \( x^{-qr} \) (note the negative exponent as it comes from the term \( b/x^{qr} \) ), we aim to set the overall exponent to zero and find when the term becomes constant (that is, not dependent on \( x \) at all). Understanding how to manipulate these exponents is vital in simplifying the term to get that constant value.

When we address the topic of Solving Exponential Equations, we're dealing with equations where variables appear in the exponents. The key to solving these equations lies in knowing how to manipulate exponents and often involves setting powers equal to each other or using logarithms to bring down exponents and solve for the unknown variable.

In the context of our binomial theorem exercise, setting the combined exponents of \( x \) to zero forms an exponential equation \( p(n-r) - qr = 0 \). Equations of this sort need to be rearranged to isolate the variable of interest—in this case, \( r \). Once the equation is set up correctly, finding the value of \( r \) becomes much more straightforward, leading us to the term in the expansion that remains constant.

The Binomial Coefficients are the numbers that serve as multipliers of the terms in the expansion of a binomial expression. They can be found in Pascal's triangle or calculated using the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \), where \( n! \) denotes the factorial of \( n \) and \( {n \choose k} \) is read as '\( n \) choose \( k \)'.

This concept is very much relevant to our exercise, as each term in the binomial expansion is prefaced with a binomial coefficient, taken here as \( ^nC_r \). It represents the number of ways to choose \( r \) elements out of \( n \) without regard to order. Understanding binomial coefficients is fundamental since they are paired with the variables and their respective powers in each term of the binomial expansion. Recognizing and calculating the correct binomial coefficient is a pivotal step in solving binomial theorem-related problems.