Api Hasthanasombat

First principle explanations

We should do more of these.

I will attempt to elaborate. My view on the benefits of explanations from first principles are several fold. It eliminates a large amount of 'clutter' that one has to store in the brain in regards to a topic, where the precious (and limited!) space can be used elsewhere, because a very large proportion of it can be derived. This also then ensures 'strong' understanding. With this understanding, it is then easier (I believe) to adapt it to new situations or to extend the concepts further.

A simple example from mathematics might help clarify what I'm trying to convey

Let's start with the first principle. The multiplication principle, this is all that really needs to be 'remembered', if we can call it such, because it is a fairly intuitive concept that comes quite naturally in itself.

If you have x ways to do something and y ways to do another thing, then there are xy ways to both things

- Wikipedia

This first principle can be used to answer lots of other problems!

How many possible permutations of n objects? First we have n choices, then n-1, n-2 ... 1, by the principle we then have n (n-1) (n-2) ... 1 possible choices. Or with modern notation, n!

What if we have n objects but we only want to pick k objects from it and put them in a line, then how many ways can we do that (Where order matters) ? First we have n choices, then n-1, n-2 ... until n-k, so everything after n-k gets 'chopped' off, or in its normally seen form, n! / (n-k)!

Ok, what if the we want the same as above, but we don't care about the order? So say we have 1,2,3,4,5 and we want to pick 3 numbers, and we regard the choice 1,2,3 the same as 2,3,1 (like picking a team of 3 from 5 people, we don't care about the order, they're the same people!) how many possible choices do we have?

So if we consider 1,2,3 and 2,3,1 the same, how many of these things that are the same are there? Well since they're n! possible ways of ordering n items we have 3! choices that are the same in this case. Since we're picking 3 objects, we can just divide the answer above (n!/(n-k)! by 3! So in general n!/(n-k!)k!

All just from knowing that if you had 2 choices for starters and 3 choices for mains you have 6 distinct choices for your whole meal!