Why are we having factorial sign when there is already the $\Pi$ notation in use?

Question

https://mathmaine.com/2010/04/01/sigma-and-pi-notation/ According to the above website the Pi notation is doing exactly what the factorial symbol does then what is the significance of Having two symbols which have the same function ? Please see that I am a 8th Grader

Answer 1

The short answer is that it makes everything shorter, easier to read, easier to write, and easier to think about. That's what good notation should do. We would much rather write

$${n \choose m} = \frac{n!}{m! (n-m)!}$$

than

$${n \choose m} = \frac{\left( \prod_{i=1}^n i \right)}{\left( \prod_{i=1}^m i \right) \left( \prod_{i=1}^{n-m} i \right)}$$

and similarly we would much rather write

$$e^x = \sum_{n \ge 0} \frac{x^n}{n!}$$

than

$$e^x = \sum_{n \ge 0} \frac{x^n}{\prod_{i=1}^n i}$$

or, God forbid,

$$e^x = \sum_{n \ge 0} \frac{\prod_{i=1}^n x}{\prod_{i=1}^n i}.$$

There is actually some interesting mathematics to talk about here. When assigning meanings to different words in a language you inevitably only have a limited supply of short words; how should their meanings be assigned? It's actually not hard to see that if there are some meanings you want to use very frequently, then to make your messages as short as possible, the shortest words should be assigned the meanings you want to use the most frequently, and you can prove a precise statement to this effect, related to Huffman coding. It's worth meditating on why the shortest words in English (or whatever other languages you may speak) have the meanings that they do: "eat," "fire," "dog," etc.

$n!$ is very short notation and that's appropriate because factorials are a concept we constantly invoke all the time in mathematics and we want to be able to do it as quickly as possible.