Talk:Probability/Combinatorics

Proof of Permutations and Combinations
How the Permutations and Combinations have been obtained should have been proved.

Repeated Section
The Counting Principle already has its own lesson. There's no need to go through a detailed explanation of this section again. I suggest we remove this section from the Combinatorics lesson. --Jimli536 (discuss • contribs) 14:37, 16 October 2016 (UTC)

Removal of introduction
I found the introduction to this book quite interesting, but fear that is will scare its intended audience, which is clearly people who want a quick and informal introduction to material covered in more depth by standard academic textbooks. Since it was interesting, I have placed it here: - Often, in experiments with finite sample spaces, the outcomes are equiprobable. In such cases, the probability of an event amounts to the number of outcomes comprising this event divided by the total number of outcomes in the sample space. While counting outcomes may appear straightforward, it is in many circumstances a daunting task. For example, consider the number of distinct subsets of the integers $$\{1,\dots,n\}$$ that do not contain two consecutive integers. This number is equal to
 * $$\frac{\phi^{n+2}-(1-\phi)^{n+2}}{\sqrt5}$$

where $$\phi=\frac{1+\sqrt5}{2}$$ is the golden ratio. It can also be obtained recursively through the Fibonacci recurrence relation.

Calculating the number of ways that certain patterns can be formed is part of the field of combinatorics. In this section, we introduce useful counting techniques that can be applied to situations pertinent to probability theory. -


 * Also: I am placing all the examples at the bottom. This is a short survey/introduction; we need to keep it concise.  Nothing except the introductory comments about the golden ratio is removed. I am just rearranging.

File:Powerset.png needs to be moved and it needs to be edited!

--Guy vandegrift (discuss • contribs) 22:34, 19 September 2021 (UTC)

Attempt to reorganize

 * Special:Permalink/3136476 (precedes my first edit)

The more I edit this, the more I see fundamental organizational issues. For reference, I show the original page in the following permalink-Guy vandegrift (discuss • contribs) 22:49, 19 September 2021 (UTC):

Removal of Power Set and Characteristic function
I may be wrong, but I see no reason for this in the "Counting Principle" section:

";Example: The power set of S, denoted by $2^S$, is the set of all subsets of $S$ . In set theory, $2^S$ represents the set of all functions $S\to\{0,1\}$ . By identifying a function in $2^S$ with the corresponding preimage of one, we obtain a bijection between $2^S$ and the subsets of $S$. In particular, each function in $2^S$ is the characteristic function of a subset of $S$.

Suppose that $S$ is finite with $n="

- S

This is far too advanced IMHO.--Guy vandegrift (discuss • contribs) 03:36, 20 September 2021 (UTC)
 * I will return this to the chapter, but now as an exercise at the bottom. With this edit, I will have restored everyting in the original artical as per Special:Permalink/3136476--Guy vandegrift (discuss • contribs) 19:43, 21 December 2021 (UTC)

Counting principle (material deleted)
After a bit of though I decided to delete to items from this section that do not really belong. The material is placed below in the event that we decide to undelete:

==The Counting Principle==

The Fundamental Rule of Counting: If a set of choices or trials, $$T_1,\dots,T_k$$, could result, respectively, in $$n_1,\dots,n_k$$ possible outcomes, the entire set of $$k$$ choices or trials has $$n_1\times\cdots\times n_k$$ possible outcomes. (The numbers $$n_1,\dots,n_k$$ cannot depend on which outcomes actually occur.)

By the Fundamental Rule of Counting, the total number of possible sequences of choices is $$5\times4\times3\times2\times1=120$$ sequences. Each sequence is called a permutation of the five items. A permutation of items is an ordering of the items. More generally, by the Fundamental Rule of Counting, in ordering $$n$$ things, there are $$n$$ choices for the first, $$n-1$$ choices for the second, etc., so the total number of ways of ordering a total of $$n$$ things is $$n\times(n-1)\times(n-2)\times\cdots\times3\times2\times1$$. This product is written $$n!$$, which is pronounced "n-factorial." By convention, $$0!=1$$

 - :The counting principle is the guiding rule for computing the number of elements in a cartesian product as well.
 * $$S\times T=\{(x,y)|x\in S\land y\in T\}$$ (STRIKE EQUATION too)

The number of elements in the cartesian product $$S\times T$$ is equal to mn. Note that the total number of outcomes does not depend on the order in which the experiments are realized.

- Consider an experiment consisting of flipping a coin and rolling a die. There are two possibilities for the coin, heads or tails, and the die has six sides. The total number of outcomes for this experiment is $$2\cdot6=12$$. That is, there are twelve outcomes for the roll of a die followed by the toss of a coin:
 * Example - Flipping a coin and rolling a die
 * $$1H,1T,2H,2T,3H,3T,4H,4T,5H,5T,6H,6T$$ .--Guy vandegrift (discuss • contribs) 01:55, 21 September 2021 (UTC)

FIGURE LABELING CONVENTION
This revised version uses so many figures that I need a convenient way to label them. Eventually they will be labeled Figure 1, Figure 2, ... But we also need a temporary system that permits us to insert a new figure between two established ones figures. We can convert between the two systems using copy/replace with minimal effort: