Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Combinations

Let $A, B, C$ be three letters , then we can combine any two of them in the following ways:
$$ AB, BC, AC$$
Similarly, if $A, B, C, D$ are four letters, then we can combine any two of them in the following manner:
$$ AB, AC, AD, BC, BD, CD$$
Similarly, we can combine any $3$ of $A, B, C, D$ as :
$$ ABC, ABD, ACD, BCD$$

We can generalize by saying that the number of all combinations of $n$ distinct things taken $r$ at a time $(r \le n)$ is

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


Note
  • In combinations, the order of the letters (or things) is not considered. Here, AB and BA are the same, so they are only counted once, unlike permutations.
  • The term "combination" is generally used for selection of things and "permutations" are used for rearrangements.

Combinations with Restrictions

  1. Number of combinations of $n$ things taken $r$ at a time in which $x$ particular things always occur is $${n-x}\choose{r-x}$$
  2. Number of combinations of $n$ things taken $r$ at a time in which $x$ particular things never occur is $${n-x}\choose{r}$$
  3. Number of ways of selecting $0$ or more things from a group of $n$ distinct things is $$2^n$$. 
  4. Number of ways of selecting one or more things from a group of $n$ distinct things is $$2^n-1$$
  5. Number of selections of $k$ consecutive things out of $n$ things in a row is $$n-k+1$$

Divisions of Identical Items into Groups

  1. The number of ways of dividing $n$ identical things among $r$ people (or groups), each of who, can receive zero or more things is $${n+r-1}\choose{r-1}$$
  2. The number of ways of dividing $n$ identical things among $r$ people, each one of whom receives at least one item is $${n-1}\choose{r-1}$$

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    Equilateral and Equiangular Polygons

    A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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    Irrational Numbers

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