Wednesday, May 13, 2009

BOB for combinatorics (=

Hey everyone!

Well I started off this unit feeling pretty comfortable with what we were doing and I thought it would be pretty easy. After the third class I started to change my mind.

Here is a brief summary of what I learned during this unit
The Fundamental Principle of Counting: This means if you have M ways of doing one thing and N ways of doing another then you have MN ways of doing both things.

An example for a question where you could apply this is, how many ways can you seat 4 students in 4 desks?

Answer: 4 x 3 x 2 x 1 = 24 ways

The first person has a choice of 4 desks to sit in, the second person has a choice of 3 desks etc..

Another way of writing this is 4! which means 4 x 3 x 2 x 1. You say this as four factorial.

Permutations: A permutation is when then order matters and there is no repition!

An example would be a question like, there are 5 horses in a race, how many ways can they finish first second and third?

You can do this in 2 ways...

Fundamental principle of counting: 5 x 4 x 3= 60
Use the pick formula: nPr= n!/ (n-r)! (do this on your calculator by hitting math, left arrow, 2)
5P3= 5!/ (5-3)!
= 120/ 2!
= 60

Permutations of Non- Distinguishable Objects (here's where i started to get a little confused)
Use this when attempting to arrange non- distinguishable objects among distinguishable ones.

An example is, how many different "words" can be made out of the letters from the word blogger?

Answer: 7!/ 2!= 2520
There are 7 letters in the word blogger and 2 of them are non- distinguishable (the 2 g's).

Circular Permutations (I do not like these they will be my downfall!!!)
This is the number of ways L objects can be arranged in a circle. To figure out these questions when there are no restrictions simply go (n-1)!

An example with restrictions is... How many different ways can 3 couples sit at a circular table if they must sit opposite each other.

BRACELETS/ NECKLACES!! (these are tricky!)

To figure out a question involving a bracelet or necklace use the formula (n-1)! / 2. We have to do this because a bracelet (or any object that can be flipped over) will have half as many combinations. Think about it carefully (:

Combinations: Order does not matter!

An example of this is... How many ways can you select 4 committee members from a group of 10 people?

Use the choose formula: nCr= n!/ (n-r)!r!
10C4= 10!/ (10-4)! 4!
= 10!/ 6! 4!
= 210

The Binomial Theorem: Use this for finding the nth term (: Remember the patterns!!

1: The coefficient of the ith term is nC (i-1)
2. The exponent on a is given by [n- (i-1)]
3. The exponent on b is given by i-1
4. exponent on a + exponent on b = n
5. The number of terms in any binomial expansion is (n+1)

Ummm so I think that mostly covers the basics... wow this turned out pretty long.. Just remember to watch out for those trick questions that seem like a lot of work but are only worth one mark...


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