combinations

Whats the Difference?

"My fruit salad is a combination of apples, grapes and bananas"

"The combination to the safe was 472"

• We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

• Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

•If the order doesn’t matter, it is a COMBINATION

•If the order does matter, it is a PERMUTATION

In Maths we use precise language..

A PERMUTATION IS AN ORDERED COMBINATION.

The ( ) Notation•Can also be written as C aswell

as nCr and C(n,r).

•It gives the number of ways of choosing r objects from n different objects.

•It is pronounced ‘n-c-r’ or ‘n-choose-r’.

nr

nr

How to Calculate It.

( )nr

= n!r! (n - r)!

( )nr

= n(n - 1)(n - 2)...(n - r +1)

r!

Definition! Practical!

You have a go!•Question 3 on your worksheet.

• Answer 15.

• And Question 4.

• (a) ( ) = 1

• (b) ( ) = 1

n0

nn

Now a twist

•Assume you have 13 soccer players and you can pick only 11 to play.

•How many ways can you choose those players - Question 5.

•You can also find it this way!

•Think of it .. every time you choose 11 you don’t choose 2!

•Thus ( ) = ( ) = 13 × 12 = 78 13 1312 2

2 × 1

•It states that ( ) = ( )

•Proof:

The Twin Rulen nr n-r

LHS = =

RHS = = n! = n! = LHS

(n - r)!(n - (n - r))! (n - r)!r!

When you have to solve equations the following are very usefull.

Equations using (n-c-r)

( ) = 1n1 ( ) = n(n - 1) = n(n - 1)n

22 × 1 2

Example•Solve for the value of the natural

number n such that ( ) = 28.n2

Solutionn(n - 1) = 28 2

n^2 - n = 282

n^2 - n = 28 -> n^2 - n - 28 = 0

(n - 8)(n + 7) = 0

n = 8 n = - 7

Reject n = - 7 is not a natural number.Therefore n = 8.