combinations
TRANSCRIPT
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.