Factorials

Overview

Engineering reference data for Factorials in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Factorial Values Table

Properties and Applications

Factorials exhibit rapid growth, approximately following Stirling's approximation for large $n$:

$n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$

Key properties and applications include:

• Combinatorics: Fundamental for counting permutations ($P(n, k) = n!/(n-k)!$) and combinations ($C(n, k) = n!/[k!(n-k)!]$).
• Series Expansions: Appear in the denominators of Taylor series, such as for the exponential function: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$.
• Gamma Function: Extends the factorial to real and complex numbers via $n! = \Gamma(n+1)$.
• Probability: Used in distributions like the Poisson and binomial distributions.

The factorial function is recursively defined for all integers $n \ge 1$ as:

$n! = n \times (n-1)!$

with the base case $0! = 1$ (as defined in the existing content).

Interactive Charts

References

• Original Source