How many ways can we arrange 3 objects A, B, and C: Using just two How many ways can we arrange 4 objects, A, B, C, & D: Using only two Using only three Keep this. We will get back to this later today!

Math I UNIT QUESTION: How do we determine the number of options if order matters? Standard: MM1D1.b Today’s Question: How can we find the number of ways to make a 9 team batting order out of 20 people? Standard: MM1D1.b.

Factorial (!) EXAMPLE with Songs ‘eight factorial’ The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial. factorial. 8! = = 40,320

Factorial Simplify each expression. a.4! b.6! c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners = = 720 = 5! = = 120

Factorial 3! = 2! = 1! = 0! = Definition: 0! Equals

The bowling league has 8 players. How many ways are there to line up the bowlers? (Answer: 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320) How can we represent this using factorials? This works well if we are using all of our objects. What if we only use a part of our objects? 8!

Try to state our data from the warm-up using a ratio of factorials: How many ways can we arrange 3 objects A, B, and C: Using just two – 6 ways Hint: 3 objects, 2 at a time How many ways can we arrange 4 objects, A, B, C, & D: Using only two – 12 ways Hint: 4 objects, 2 at a time Using only three – 24 ways Hint: 4 objects, 3 at a time

How many ways can we pick three people from a group of 12 if order matters? (Answer: 12 * 11 * 10 = 1320) How can we represent this by using factorials? (Hint: 12 objects taken 3 at a time) 12! / 9!

Permutations The arrangement of elements in a distinct order is called a permutation. The number of permutations on n objects, taken r at a time is: The previous example would be

You are selecting a 9 person baseball team out of 12 students and making the batting order. Does order matter? YES! – therefore it is a permutation problem

Distinguishable Permutations How many distinguishable ways can the letters MOO be arranged? 3! Is not the right answer, because the two O’s look the same Make a tree diagram The answer is 3 How do we show this as factorial numbers? 3!/2!

Distinguishable Permutations How many distinguishable ways can the letters WOOF be organized? Again, 4! Is not right answer. Make a tree diagram The answer is 12 How do we show this as a factorial? 4!/2!

Distinguishable Permutations How many distinguishable ways can 2 similar marbles and three similar blocks be arranged? There are 5 objects. How many ways can 5 objects be arranged? 5! is too big because the marbles are not distinguishable, and the 3 blocks are not distinguishable. Make a tree diagram

Distinguishable Permutations The answer is 10 How do we show this as factorial numbers? 5!/(3!2!) Distinguishable Permutations

Permutation Review The number of unique ways of organizing 6 unique things is 6! = 720 ways The number of unique ways of organizing 6 unique things in groups of 4 is 6!/2!, or The number of unique ways of organizing 6 things, but 4 are the same and 2 are the same is:

Class Work Page 344, # 1 – 28 all, and find the number of unique permutations for the letters of the following words: mississippi armageddon supercalifragilisticexpialidocious