Assignment 16 (2A.2.3) Homework: Pg.46 (5-10)

Warm Up Factor (take out the greatest common factor first) a. 3𝑥! + 24 b. 12𝑥! − 75𝑦! c. 32𝑥! + 18

Notes Factorials (n!) tell you to multiply all the descending integers from n. examples: 5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 3! = 3 ∙ 2 ∙ 1 0! = 1 Counting Principles: Permutations are arrangements of items in a particular order. nPr =

!!!!! !

gives you the number of ways you arrange r items from a set of n.

example: If there are 8 different countries in the final Olympic event, then there are 8P3 ways the medals can be awarded. 8P3 = !!

!!! != !∙!∙!∙!∙!∙!∙!∙!

!∙!∙!∙!∙!= 8 ∙ 7 ∙ 6 = 336  

Combinations are arrangements of items which order does not matter. nCr = !!

!! !!! !   gives you the number of ways you can group

r items from a set of n. example: If there are 38 students in the class, then there are 38C4 ways to make groups of 4. 38C4 = !"!

!!!"!= !"∙!"∙!"∙!"

!∙!∙!∙!= 73,815

Ex.1 Compute a. 7P4 b. 16P3 c. 7C4 d. 13C8 Ex.2 Pizza Combinations If there are 20 possible toppings you can put on a pizza, how many different 3 topping pizzas can you make? 5 toppings? Ex.3 Maroon 5 Concert a. You won five tickets to the Maroon 5 concert and now have the burden of asking only four other friends. If you have 10 really good friends, how many different combinations of friends can you bring to the concert? b. You decide to rank your friends to help you pick who to bring to concert. How many different ways can you rank your top 4 (out of 10) friends? Finding a specific term of a binomial expansion:

The kth term: nCk-1 𝑎!!(!!!)𝑏!!! Ex.4 Find the specified term in the binomial a. the third term of 𝑥 + 𝑦 ! b. the eight term of 𝑥 − 2𝑦 !" c. the seventh term of 𝑥! + 3𝑦 !! d. the fifth term of 𝑥 − 3 !"

Looking for Patterns

(𝑎 + 𝑏)!= nC0 𝑎! + nC1 𝑎!!!𝑏 + nC2 𝑎!!!𝑏! + nC3 𝑎!!!𝑏! + … + nCn 𝑏!