On the relationship between exponents and factorials

Through the process of reducing the product of the numbers 10-1 raised to the 2nd power, 3rd power, 4th power, and 5th power, I have come across a pattern which reflects the result of factorials!

[At this time, please do review my work in the attached excel documents]

In this way, I have created a "formula" that requires professional revision. To be specific, I want to know how these two form of mathematics would be applied to a problem, and what type of problems would favor one over the other - and vice versa.

• It's quite tricky to go through the table and to try to see exactly what you have done and what the pattern is. I would recommend you approach the question algebraicly, by looking at numbers of the form $b^n$, where $b$ is the base and $n$ is the exponent. You are then looking at the numbers of the form $b^n - (b-1)^n$ etc ... If you write your expressions on paper like this, perhaps you'll get some insight to the pattern that you have discovered, and why such a pattern occurs. – Matti P. Oct 26 '18 at 6:16
• Possible duplicate of Repeatedly taking differences on a polynomial yields the factorial of its degree? – Chris Culter Oct 26 '18 at 6:17
• See in particular user21820's answer, which speaks to applications: math.stackexchange.com/a/2319671/87023 – Chris Culter Oct 26 '18 at 6:18