1. Chapter10.1-10.3

2.  Determine how many different possibilities are possible:  1. There are 3 different ice cream flavors and 5 different toppings. You can have one type of ice cream and one topping.  2. You have 30 different shirts, 8 types of pants, and 4 different types of shoes. How many different ways can you dress yourself?  3. You have just enough money to go out to eat and see a movie. There are 5 different restaurants near the movie theater and 10 different movies playing.

3.  If three events occur in m, n, and p ways, then the number of ways that all three events can occur is m x n x p.  It can be ANY number of events.

4.  a. repetition is allowed  b. repetition is not allowed.  1. A 4-digit lock with numbers 0-9.  2. A 6-digit lottery with numbers from 1-30.  3. A license plate with 3 letters followed by 4 numbers.

5.  How many ways can you pick r things out of n, where ORDER MATTERS.  n P r =  You can plug these into your calculator. MATH  PRB  nPr.

6.  First, use the Fundamental Counting Principle.  Then, use the Permutations Formula by hand.  1. A TV news director has 8 news stories to present on the evening news.  a. How many different ways can the stories be presented?  b. If only 3 stories will be presented, how many possible ways can a lead story, a second story, and a closing story be presented?

7.  First, use the Fundamental Counting Principle.  Then, use the Permutations Formula by hand.  2. 10 students at Norwin are running for President.  a. How many different ways can the students give their speeches to the school?  b. First place becomes President, second place becomes Vice-President, third place becomes Treasurer, and fourth place becomes Secretary. How many ways can the students be P, VP, T, and S?

8.  How many different permutations can you make with the following letters:  1. ABCD  2. ABCC  3. ABBB

9.  different permutations of n objects where one object is repeated s 1 times, another repeated s 2 times, and so on is:

10.  1. KAYAK  2. TALLAHASSEE  3. CINCINNATI

11.  10.1 #11-16, 32-53x3, 64-66

12.  Place 44, 50, and 64-66 on the board. Show your work!  At your seats, answer the following questions:  1. How many different ways can Ms. Rothrauff call on students to write the above answers on the board?  2. How many different ways can you pick 4 lunch sides given that there are 10 options?  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

13.  How many different permutations can you make with the following letters:  1. ABCD  2. ABCC  3. ABBB  How does this help prove this is true?

14.  The definition of a factorial is n!=n x (n-1)!  Use this information to prove that 0!=1.

15.  Discuss in your groups what you think the formula will be for Combinations (where order DOES NOT MATTER).  Consider the following: ◦ Permutation Formula from yesterday. ◦ Different Permutations of ABCD picking all 4 letters. ◦ If order DID NOT MATTER, how many different possibilities would there be to order ABCD using all 4 letters?

16.  How many ways can you pick r things out of n, where order DOES NOT MATTER.  n C r =  You can plug these into your calculator. MATH  PRB  nCr.

17.  When finding the number of ways both event A AND event B can occur, you need to multiply.  When finding the number of ways that event A OR event B can occur, you add instead.  Pg 691

18.  Counting problems that involve phrases like “at least” or “at most” are sometimes easier to solve by subtracting possibilities you do not want from the total number of possibilities.  Pg 691

19.  The Norwin Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshman.  a. How many different committees of exactly 2 seniors and 2 juniors can be chosen?  b. How many different committees of at most 4 students can be chosen?

20.  You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?

21.  In a standard deck of 52 cards:  1. How many ways can you get a flush in hearts?  2. How many ways can you get all red cards?  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).  *flush=all same suit (hearts, diamonds, etc.)

22.  3. How many ways can you get at most one heart?  4. How many ways can you get at least one 6?  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

23.  10.2 #3-10, 13-18

24.  Place numbers 14, 16, and 18 on the board. Show your work!  At your seats, answer the following question:  How many ways can you get a full house with a standard deck of 52 cards?  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).  *full house=3 of the same type and 2 of the same type (QQQKK, 444JJ, 33399, etc.)

25.  Refer to page 692 in your books.  If you arrange the values of n C r in triangular pattern in which each row corresponds to a value of n, you get Pascal’s Triangle.  The r corresponds to the number in that row.  *You start counting with 0. Both the rows and the number in that row.*  * 0 C 0 = 1 and is the 0 th row.*

26.  1. From a collection of 7 baseball caps, you want to trade 3. Use Pascal’s Triangle to find the number of combinations of 3 caps that can be traded.  2. The 7 members of the math club chose 2 members to represent them at a meeting. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives.

27.  Refer to page 693 in your books.  Steps to use the Binomial Theorem:  1. Identify a, b, and n.  2. Make a list of all the C terms vertically. n=n for all C terms, while r starts at 0 at the top and goes to n on the bottom. (There will be n+1 C terms.)  3. Next to each C term, write the a term in parenthesis. Raise each a term starting at the top to the nth power down to the bottom ending with 0 th power.  4. Next to the a term, write the b term in parenthesis. Raise each b term starting at the top to the 0 th power down to the bottom ending with the nth power.  5. Multiply all of the terms out and put a “+” between each new term.

28.  1. (x+y) 6  2. (5-2y) 3  3. (3x-2) 4

29.  Find the coefficient of x r in the expansion of (a+b) n.  Formula: n C r a r b n-r

30.  1. Find the coefficient of x 5 in the expansion of (x-3) 7.  2. Find the coefficient of y 3 in the expansion of (5+2y) 8.  3. Find the coefficient of x 3 y 4 in the expansion of (2x-y) 7.

31.  10.2 #19-33odd, 38-39, 48-49

32.  Place 25 and 31 on the board. Show your work!!!  At your seats, try 24 and 26 on page 695.

33.  1. How many different possibilities are there to win a lottery if 3 numbers are drawn from 1-15…  a. With repetition?  b. Without repetition?  2. What would be the probability of winning the lottery…  a. With repetition?  b. Without repetition?

34.  Theoretical Probability of event A:  P(A)=  Experimental Probability of event A:  P(A)=

35.  You pick a card from a standard deck of 52 cards. Find the following probabilities:  1. Picking an heart.  2. Picking a red King.  3. Picking anything but an Ace.  4. Picking a number card (2-9).  5. Picking a Joker.  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

36.  You have an equally likely chance of picking any integer from 1-20. Find the probabilities:  1. Picking a perfect square.  2. Picking a factor of 30.  3. Picking a multiple of 3.

37.  Odds in favor of event A=  Odds against event A=

38.  You pick a card from a standard deck of 52 cards. Find the following odds:  1. Odds in favor of drawing a 5.  2. Odds against drawing a diamond.  3. Odds in favor of drawing a heart.  4. Odds against drawing a Queen.  Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

39.  You throw a dart at the board. Your dart is equally likely to hit any point inside of the board. What is the probability of getting 0 points? What is the probability of getting 50 points? Are you more likely to get 0 points or 50 points? 3in 3in 3in 0pts 50pts

40.  10.3 #4-18even, 20-23, 35-39