Factorials

Example

Fred knows that the code to his locker is made up of the digits 5,6, 7 and 8 , but he can’t remember the order. How many different  possible codes exist to open his locker ?

There are 4 ways of listing the first number,

5***           6***                   7***              8***

and 3 ways of listing the second number.

56**	65**	75**	85**
57**	67**	76**	86**
58**	68**	78**	87**

This leaves 2 ways of selecting the third number

567*	657*	756*	856*
568*	658*	758*	857*
576*	675*	765*	865*
578*	678*	768*	867*
586*	685*	785*	875*
587*	687*	786*	876*

Finally, this leaves only one way of selecting the last number.

5678      6578     7568       8567
5687      6587     7586       8576
5768      6758     7658        8657
5786      6785      7685       8675
5867      6857      7856       8756
5876     6875      7865        8765

 

There are 24 possible codes.

4x3x2x1=24
This is called  4-factorial and is written 4!

0! = 1  By definition
1! = 1
2! = 2X1=2
3! = 3X2X1 = 6
4! = 4X3X2X1=24
5! = 5X4X3X2X1 = 120

Note,
5 x 4! = 5x 24 = 120 = 5!

Permutations and Combinations

A permutation is an ordered arrangement  of objects, in which order matters.

A group of n different objects has  n!  possible  permutations .

A combination is  an arrangement  of objects where order does not matter.

Example

There are 56 ways of choosing 5 different tins of catfood from  8 brands, but 6720 ways of ordering them!

Properties of combinations

 

Pascal's triangle

 

The triangle is built by adding the row above.

 

 

This can be written ⁿC_r   , where n is the row, r is the column

Notice how³C₃=³C₀ = 1 

 

and ³C₂=³C₁ = 3 

 

and ⁴C₃=⁴C₁= 4 

 

Also, notice

and

 

 

 

 

The coefficients are known as binomial coefficients.

ⁿC_r can be written in the form

So for the example given above

 

Properties to learn

 

 

 

 

Equations

Example

Solve the equation

From the properties above, since

 

Solution

 

Example

Solve the equation

 

Binomial expansions

Example

Notice that the powers of x and y both add up to 5, and that as the powers of x decrease from 5 to 0, the powers of y increase from 0 to 5.

In General

 

This is known as the binomial theorem, and gives the expansion of (a + b)ⁿ , where a and b are real numbers and n is a natural number.

The binomial coefficients are found in the n^th row of Pascal’s triangle.

Examples

Expand the following:
     

       

 

The binomial theorem allows a specific term to be found from the  general form.

      

     

Example
Find the seventh term in the expansion of (2x +3y)⁹

     

Find the term containing x³ in the expansion of (3 +2x)⁵

          

Probability and the binomial theorem

   

Example

 

The binomial expansion and e

 

Examples