Solving a Complex Equation: A Deep Dive into Combinatorial Mathematics

Mathematics is a vast and intricate field, encompassing a multitude of equations and theorems that challenge our understanding of numbers and their relationships. One such equation, although relatively complex, illuminates the fascinating world of combinatorial mathematics and number theory. Let's delve into the details of the equation:

Equation Breakdown

The given equation is:

$binom{n 2}{4} n^2 - 3$

First, we rewrite the left-hand side:

$frac{(n 2)(n 1)n(n-1)}{24} n^2 - 3$

Manipulating the Equation

By manipulating both sides of the equation, we can derive a more manageable form:

$(n^2 n - 2) (n^2 n) 24n^2 - 72$

Further simplification leads us to:

$(n^2 n - 1)^2 24n^2 - 71$

Since squares of real numbers are non-negative, we derive:

$n geq 2$

Further Expansion and Simplification

To simplify further, let's expand the left-hand side:

$(n^2 n)^2 - 2(n^2 n) 24n^2 - 72$

Expanding and simplifying this equation, we get:

$n^4 2n^3 - 25n^2 - 2n 72 0$

Testing Integer Solutions

We begin by testing integer values to see if they satisfy the equation. Let's start with (n 2):

$1 cdot 16 2 cdot 8 - 25 cdot 4 - 2 cdot 2 72 16 16 - 100 - 4 72 104 - 104 0$

Thus, (n 2) works as a solution. We then factorize the equation:

$(n - 2)(n^3 4n^2 - 17n - 36) 0$

Now, we test the cubic expression for (n geq 4):

$(n 1)^3 4(n 1)^2 - 17(n 1) - 36 - (n^3 4n^2 - 17n - 36) 3n^2 11n - 12 geq 3 cdot 16 11 cdot 4 - 12 48 44 - 12 80 > 0$

Conclusion

Therefore, the only non-negative integer solution to the original equation is (n 2). This highlights the importance of combinatorial numbers in mathematics, which are conventionally defined exclusively for non-negative integers.