Factorials

 

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

 

In general

n! = nx(n-1)x(n-2)x(n-3)………x3 x 2 x1

 

Note,

5 x 4! = 5x 24 = 120 = 5!

 

 n x (n-1)! = n!     so    (n-r+1)x (n-r)! = (n-r+1)!     

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.

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

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

 

                                      1

1        2  1

                              1   3   3   1

                           1   4    6   4   1

                        1   5   10    10  5  1

                      1  6  15    20    15  6  1

                    1  7  21  35    35    21  7  1

                  1  8  28  56  70   56   28  8   1

                1  9  36  84  126  126  84  36 9 1

             1  10 45  120 210 252 210 120 45 10 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                column 0   column 1   column 2   column 3   column 4   column 5

row 0        1                                                                             

row 1        1              1              0              0              0              0

row 2        1              2              1              0              0              0

row 3        1              3              3              1              0              0

row 4        1              4              6              4              1              0

row 5        1              5              10             10             5              1

 

Can be written nCr   , where n is the row, r is the column

 

                column 0   column 1   column 2   column 3   column 4   column 5

row 0        0C0                                                                           

row 1        1C0            1C1            0              0              0              0

row 2        2C0            2C1            2C2            0              0              0

row 3        3C0            3C1            3C2            3C3            0              0

row 4        4C0            4C1            4C2            4C3            4C4            0

row 5        5C0            5C1            5C2            5C3            5C4            5C5

 

Notice how 3C3=3C0 = 1  and 3C2=3C1 = 3 

The coefficients are known as binomial coefficients.

 

Also, notice

 

 

 

 

nCr can be written in the form

 

so

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solve the equations :

 

 

 

 

 

 

 

 

Binomial expansions

 

 

 

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 nth 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

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

The binomial expansion and e