Binomial Theorem

What up G's ,

A couple days ago we have went over Pascal's Triangle and Binomial Theorem.
Pascal's Triangle is a table of  Binomial Coefficients.

<--------- (x + y)0   =                    1
<--------- (x + y)1   =                1x+1y

<----------(x + y)²   =           1x + 2xy +1y
<----------(x + y)3   =     1x + 3x y + 3x y + 1y
<----------(x + y)4   = 1x + 4xy + 6xy + 3xy + 1y



Binomial Theorem is a formula to quickly expand binomial expressions to locate a specific term.

Formula: Tk+1=nCk A^n-k B^k

Example 1                  Using the formula to find the 4th term of (x-2y)^10

K= 4-1 = 3                               Tk+1=nCk A^n-k B^k

N= 10                                        T3+1=10C3 (x)^10-3 (-2y)^3

A= X

B= -2y                                      T4= 120 * x^7 * -8y^3

T4= -960x^7y^3

COMBINATIONS

Good greetings, friends and David :)

In class we have been studying about combinations. First things first you must become a expert on telling the difference between a permutation and a combination. The difference between permutations and combinations is:

         Permutation                                   Combinations
                 2 Actions:                                        1 Action:
                          1. Select                                           1. Select
                          2. Arrange                     *No dash method for combinations*
             
To tell the difference you must always ask yourself: "is the order important or not?".

Example 1: "How many can you select 2 fruits to eat if there are 12 to choose from"
- We are selecting 2 fruits from 12 options
- The order does not matter
- Therefore it is a combination

Example 2:"The combination to the safe is 472". 
- Now we do care about the order. "724" won't work, nor will "247". It has to be exactly 4-7-2.

Formula:

Example: Evaluate:

10C2

n = 10                      n! / r!(n-r)!                 

r = 2                        10! / 2!(10-2)!
                                10! / 2!(8)!        =  45

Permutations

Hey Fam

So in class we started taking a look at permutations and how we can use it to help us solve problems that we may have that involve arrangement. A Permutation is to arrange or rearrange a set or collection of objects in a certain, specific order without repetition, unless stated.

We can solve permutation problems by using the dashed method, or the formula

      Formula: nPr = n!/(n-r)!
      "n" is equal to the total number of objects and entities
      while "r" is equal the amount of objects you pick

      This means that "n" must always be greater than "r"

Example

Question: A restaurant has 8 available tables for customers. If 5 different groups of people enter the restaurant, how many different ways can they be seated in the restaurant?

Answer:            nPr = n!/(n-r)!                                                   8P5 = 8!/(8-5)!

                          n = 8                                                                         = 8!/3!

                          r = 5                                                                          = 6720 different ways

Hey Friends!

     So far in class we've been going over the Fundamental Counting Principal and Permutations! 
  we went through various methods to do certain problems. I've scrounged up a couple videos to better our understanding. Please note that these are things we've all seen in class.

Fundamental Counting Principal (Stop at 1:30)

P.S. do i get candies for this???

Welcome

Welcome to our blog. This space is designed for students of the Maples Collegiate, attending the Pre-Calculus 40S class, section C, with Mr.P. We are going to use this space to discuss our daily lessons, ask questions you didn't get a chance to ask in class, and to share your knowledge with other students. Most importantly we will use this blog to reflect on what we're learning.
Have a great semester.