The sum of all possible products of the first n natural...

The sum of all possible products of the first $n$ natural numbers taken two at a time is (A) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma n\right]$ (B) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n\right]$ (C) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma(n+1)\right]$ (D) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n^{2}\right]$

The sum of all possible products of the first $n$ natural numbers taken two at a time is
(A) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma n\right]$
(B) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n\right]$
(C) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma(n+1)\right]$
(D) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n^{2}\right]$

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Key Concepts

Binomial Coefficient

This concept involves selecting a fixed number of elements (in this case, two) from a larger set, without regard to the order of selection. It is a fundamental principle in combinatorics, often denoted as C(n, 2) or 'n choose 2', which counts the number of unique pairs that can be formed from n elements.

Square of Sum Identity

This algebraic identity states that the square of the sum of a set of numbers can be expressed as the sum of their squares plus twice the sum of all distinct products of pairs of different numbers. This identity is crucial for transforming and simplifying expressions involving pairwise products into a form that involves the entire sum and the sum of the squares.

Series Summation Formulas

These formulas provide a method to compute sums of sequences, such as the sum of the first n terms of an arithmetic series or the sum of squares of the first n numbers. They are essential tools in algebra and number theory, allowing for the derivation and simplification of expressions that include these series in closed form.

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