The traditional Binomial Theorem is specifically designed for binomials, which are expressions with two terms. However, the concept can be extended to polynomials with more than two terms using the Multinomial Theorem. The Multinomial Theorem allows for the expansion of a polynomial expression (a + b + c + ...)ⁿ and involves multinomial coefficients, which are generalisations of binomial coefficients. The coefficients and terms in the expansion become more complex as they involve multiple variables and their respective powers. While the Binomial Theorem provides a straightforward approach for two-term expressions, the Multinomial Theorem caters to the expansion of expressions with three or more terms.

The Binomial Theorem and the Fibonacci Sequence are related through a fascinating connection with Pascal's Triangle. If you sum up the entries of the diagonal elements starting from the leftmost element of each row in Pascal's Triangle, you obtain the Fibonacci Sequence. Mathematically, this can be expressed as F(n) = C(n, 0) + C(n-1, 1) + C(n-2, 2) + ... + C(1, n-1) + C(0, n), where F(n) is the nth Fibonacci number and C(n, k) represents the binomial coefficient. This relationship beautifully illustrates how two seemingly distinct mathematical concepts, the Binomial Theorem (via Pascal's Triangle) and the Fibonacci Sequence, intersect and enrich our understanding of numerical patterns and structures.

The Binomial Theorem can be employed to swiftly calculate powers of numbers, especially when dealing with expressions in the form of (a + b)ⁿ. By using the theorem, you can expand the expression without having to multiply it out the long way. For instance, to calculate 1.05¹⁰ quickly, you can rewrite 1.05 as (1 + 0.05) and use the Binomial Theorem to expand (1 + 0.05)¹⁰. The expansion will involve terms like C(10, k) * 1^(10-k) * 0.05^k, which are significantly easier and quicker to calculate than performing the multiplication 1.05 * 1.05 * ... ten times. This method is particularly useful for approximating powers of numbers close to a base number.

The Binomial Theorem is deeply intertwined with combinatorics and probability theory, particularly in calculating combinations and determining probabilities in binomial experiments. In combinatorics, the coefficients in the binomial expansion, also known as binomial coefficients, represent the number of ways to choose k items from a set of n (denoted as "n choose k" or C(n, k)). In probability theory, especially in binomial distributions, these coefficients are used to find the probability of obtaining exactly k successes in n independent Bernoulli trials. The theorem provides a systematic way to expand the powers of binomial expressions, which is crucial in determining specific terms in probability distributions and combinatorial problems, thereby forming a bridge between algebraic expansions and discrete mathematics.

In the binomial coefficient, denoted as C(n, k) or "n choose k", n is always larger than or equal to k because it represents the total number of items to choose from, while k represents the number of items to choose. Mathematically, the binomial coefficient is defined as C(n, k) = n! / (k!(n-k)!). If k were larger than n, the factorial in the denominator (n-k)! would involve taking the factorial of a negative number, which is undefined. Furthermore, from a combinatorial perspective, it is impossible to choose k items from a set of n if k > n, as there aren’t enough items to choose from. Thus, ensuring n ≥ k maintains the mathematical and logical validity of the binomial coefficient.