If the sum of co-efficient of expansion \(\left(m^{2} x^{2}+2 m x+1\right)^{31}\) is zero then \(\mathrm{m}=\) (a) 1 (b) - 1 (c) 2 (d) \(-2\)

The expansion of a binomial expression is a fundamental concept in algebra that arises from the binomial theorem. When we talk about a binomial, we are referring to a mathematical expression that contains two terms, such as \(a + b\). The binomial theorem provides a way to expand expressions raised to a power, that is, \( (a + b)^n \), where \( n \) is a non-negative integer.

The theorem states that the expansion of a binomial expression raised to the nth power is the sum of terms involving the coefficients \( \binom{n}{k} \), the first term \(a\) raised to the power of \(n-k\), and the second term \(b\) raised to the power of \(k\), where \(k\) varies from 0 to \(n\). Mathematically, it is expressed as:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

• This expansion leads to a sequence of terms that are often referred to as 'binomial coefficients.' These are the numbers \( \binom{n}{k} \), which are found in Pascal's triangle.
• The importance of understanding the expansion lies not just in computing the power of a binomial expression, but also in various applications across mathematics and science, including probability and combinatorics.

When we expand a binomial expression using the binomial theorem, the coefficients of the resulting terms play a significant role. These coefficients, also known as binomial coefficients, are the multipliers of the terms in the expanded form. In the expression \( (a + b)^n \), the coefficient of the term involving \(a^{n-k} b^k\) is \( \binom{n}{k} \).

\[ (a + b)^n = a^n + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + ... + b^n \]

The sum of the coefficients in a binomial expansion can be found by setting both \(a\) and \(b\) to 1. What results is an expression involving sums of binomial coefficients, which always equals \(2^n\) due to the property of binomial coefficients that relates to permutation and combination formulas:

\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
Therefore, any sum of the coefficients in the expansion \( (a + b)^n \) will be positive and non-zero. This understanding is crucial to solving problems that involve evaluating the sum of coefficients under certain conditions.

A binomial coefficient \( \binom{n}{k} \) is an integral part of the binomial theorem that appears as the coefficient of the terms in the expansion. It represents the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order, which is a concept from combinatorics.

The formula to compute a binomial coefficient is:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \( ! \) denotes factorial, the product of all positive integers up to that number. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).

This means if we want to find out how many different ways we can select \(k\) items from a set of \(n\) items, we would use the binomial coefficient. Binomial coefficients also have some interesting properties:

• Symmetry: \( \binom{n}{k} = \binom{n}{n-k} \) which reflects Pascal's triangle's symmetry.
• Each term in the expansion is a series product involving these coefficients as multipliers, which further emphasizes their central role in the binomial theorem.

The concept of the binomial coefficient is not only crucial for understanding the binomial theorem but also serves as a bridge to other areas of mathematics such as probability theory and algebra.