If \(\left(1 /{ }^{4} \mathrm{C}_{\mathrm{n}}\right)=\left(1 /{ }^{5} \mathrm{C}_{\mathrm{n}}\right)+\left(1 /{ }^{6} \mathrm{C}_{\mathrm{n}}\right)\) then value of \(\mathrm{n}\) is (a) 3 (b) 4 (c) 0 (d) none of these

Combinatorial identities are equations that hold true for all values within the scope of the variables. These identities encompass relationships between various combinatorial constructs, such as binomial coefficients and permutations. An example of a combinatorial identity is the Pascal's identity, which relates to binomial coefficients:
\[ {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k} \]
This identity shows that to choose k elements out of a set of n elements is equivalent to the choice of picking one element and then choosing k-1 out of the remaining n-1, or not picking that element and straight away choosing k out of the n-1. Another combinatorial identity involves the sum of binomial coefficients on the same row of Pascal's triangle, which sums to a power of 2:
\[ \sum_{k=0}^{n} {n \choose k} = 2^n \].
These identities are not just mathematical curiosities—they have practical applications in areas such as probability, statistics, and algorithm design.

Understanding the Basics of Binomial Coefficients

Binomial coefficients, noted by the symbol \( {n \choose k} \), represent the number of ways to choose k items from a set of n distinct items where the order of selection does not matter. They are a fundamental part of combinatorics and appear in the binomial theorem, which states:
\[ (x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \]
Calculating binomial coefficients can be done through the formula:
\[ {n \choose k} = \frac{n!}{k!(n-k)!} \]
where \(!\) denotes factorial, the product of all positive integers up to a given number. This concept also directly ties in with Pascal's Triangle, a geometric representation where each number is the sum of the two numbers directly above it. Understanding binomial coefficients is essential for solving a broad range of problems in mathematics, including those concerning probability and combinations.

Algebraic Manipulation of Combinatorial Equations

The incorporation of algebraic equations in combinatorial problems allows us to solve for unknown quantities by venerating relationships between the combinatorial elements. Typically, these equations might involve binomial coefficients or other combinatorial figures and require the solver to manipulate the expression using algebraic techniques.
In our textbook problem, an algebraic equation involving binomial coefficients is used to find the value of n. The equation is manipulated by finding a common denominator to merge terms and simplify the expression. This exercise showcases how essential algebra is in solving combinatorial problems, such as proving identities or calculating the number of combinations. Mastery of algebraic manipulation when dealing with combinatorial terms is indispensable and adds another layer of understanding to the foundations of combinatorics.