# User:Saul/algebra

## Contents

## Combinations and Permutations

### Permutations/Pick

A permutation is a sequence where the order does matter.

This is often notated by **n pick r** where **n** is the data set - say 40 lotto balls and **r** is the selection count - say 8 balls.

The general formulae for permutations is:

**P = n! / (n-k)!**

### Combinations/Choose

A combination is a sequence where the order does NOT matter.

This is often notated by **n choose r** - **n** and **r** mean the same thing as in pick.

The general formulae for combinations is:

**C = n! / ((n-k)!k!)**

Also note that:

**n choose k** is the same as **n choose (n - k)**

## Binomial Theorem

**(a + b) ^{n} = _{k=0}Σ^{n} ( ( n choose k ) a^{k} b^{n-k} )**

### Approximation Using Binomial

Calculating a number to the power of another can be approximated by the following formulae:

**(1 + x) ^{n} = 1 + nx + (n(n-1)x^{2})/2! + (n(n-1)(n-2)x^{3})/3! + (n(n-1)(n-2)(n-3)x^{4})/4! ...**

Note: This is unending so the more steps you take the more accurate the number will be, however usually 3-4 steps is enough.

## Trigonometry

**Sin(30) = (√1) / 2 = Cos(60)**

**Sin(45) = (√2) / 2 = Cos(45)**

**Sin(60) = (√3) / 2 = Cos(30)**

### Various Formulae

**Sin ^{2}(x) = Sin(x) * sin(x)**

**Sin(x**

^{2}) = Sin(x * x)**Tan(x) = Sin(x) / Cos(x)**

**Sin**

^{2}(x) + Cos^{2}(x) = 1**Sin(x) = Cos(90 - x)**

**Cos(x) = Sin(90 - x)**

**A / Sin(a) = C / Sin(c) = C / Sin(c)**

**a**

^{2}= b^{2}+ c^{2}- 2bc * Cos (A)Note:

**± = ± → + = + → - = -**

**± = ∓; → + = - → - = +**

**Sin(a ± b) = Sin(a) * Cos(b) ± Cos(a) * Sin(b)**

**Sin(a ± b) = Cos(a) * Cos(b) ∓ Sin(a) * Sin(b)**

**Sin(2a) = 2Sin(a) * Cos(a)**

**Cos(2a) = Cos**

^{2}(a) - Sin^{2}(a)**Sin**

^{2}(a) = ( 1 - Cos(2a) ) / 2**Cos**

^{2}(a) = ( 1 + Cos(2a) ) / 2**Csc(x) = 1 / Sin(x)**

**Sec(x) = 1 / Cos(x)**

**Sec**

^{2}(x) = 1 / Cos^{2}(x)## Quadratics

### Turning Point

The equation dor the turning point **x** value is: **-b / 2a**, after that y can be derived.

### Complete the Square

The completed square looks like this:

**y = a(x + dx) ^{2} + dy**

**a**is the shape, a negative value indicates a

**n**like shape where a positive value indicates a

**u**like shape and if a is 0 the shape is a flat line.

**dx**is the

**negative**displacement of

**x**.

**dy**is the displacement of

**y**.

#### Formulae

To convert a general quadratic to the completed square, use the following formulae:

**y = ax ^{2} + bx + c → a(x + b / 2a)^{2} - b^{2} / 4a + c**