UNit 2 : Combinations and Binomial Theorem

1. UNit 2 : Combinations and Binomial Theorem By Seher Ali, Ramanpreet Dhaliwal, Eshita Sikder, Kiranprit Singh

2. 5.1 Organized Counting with Venn Diagrams • Venn Diagram: Used when order of the items does not matter • A set is a collection of distinct objects, denoted by an uppercase letter • The objects in a set are called elements ( or members of the set)

3. Finite Set Ex: Set A Ex: Set, Colours of the rainbow A={2,4,6,8} x={R,O,Y,G,B,I,V} • Infinite Set Ex: Set of all even positive integers E={2,4,6,8,10,...} *Remember that the order of the sets DO NOT matter.

4. Cardinality (size) of a set: Number of elements in a set n(A) = # of elements in set A Ex: Set A A= {2,4,6,8} n(A)=4 Ex: Even positive integers E={2,4,6,8,10,...} n(E) = ∞ • An empty set is a set with no elements {} – zero elements

5. Set Equality: -two sets A and B are equal if and only if they have the same elements • Subsets-sets within sets: -if all the elements of set B are also elements of set A, then B is a subset of set A Ex: A={a,b,c,d,e,f} B={c,d,f} C={d,e,f,g} Then B A C A ∩ ─ ─ ∩ ─

6. Venn Diagrams • Used to organize info and illustrate the relationship between sets and their subsets • S = Universal Set • Union of two sets – AUB = union of A and B • Intersection of two sets – A∩B= intersection of A and B

7. Formula for two Sets n (AUB) = n(A)+n(B)-n(A∩B)

8. Example

9. Formula for three Sets n(AUBUC)=n(A)+n(B)+n(C)-n(A∩B)-n(A∩C)-n(B∩C)+n(A∩B∩C)

10. Example • 114 eat meat • 100 eat bread • 70 eat fruit • 48 eat meat & bread • 41 eat meat & fruit • 27 eat bread & fruit • 17 eat all three n(AUBUC)=?

11. 5.2 Combinations • Permutations- a selection from a group of items where the order of items matter • Combination- a selection from a group of items where the order of items does NOT matter • The number of combinations of r objects chosen from a set of n distinct objects ( order doesn't matter) • nCr = n! = C(n,r) (n-r)!r!

12. Example #1 • The school's swim team (shssst) has 6 females and 5 males. For the upcoming ROPSAA swim tournament, in how many ways can: • a) a group of 4 swimmers be selected? • b) 3 males AND 4 females be chosen? • c) 3 males OR 3 females be chosen?

13. Example #2 • A used car dealer has 40 pre-owned cars for sale. However, the sales showroom only has room for 24 cars. The sales manager has been instructed to put at least 20 of the cars on display. In how many different ways can the manager select the cars to display?

14. 5.3 Problem Solving with Combinations • In general, a combination is a subset of a group of n distinct objects. • So, the total number of combinations containing at least one item chosen from a group of n distinct items is 2 -1. (remember, the 1 is if you don’t chose anything at all). n

16. Example • Alex finds 9 shirts in his size at a sale. How many different purchases can he make?

17. Combinations of items in which some are alike If at least one item is chosen, the total number of selections that can be made from p items, q items of another, r items of another kind, and so on, is (p+1)(q+1)(r+1)…-1

18. Example • How many different sums of money can be made with one $10 bill, two $20 bills, and one $50 bill?

19. A team of 9 young hockey players is to be drafted from a team of 25. • How many teams can be selected overall? • Branden, Nick, Tyler and Ryan are four of the players in the class that can be drafted. How many of the teams will include these 4 players?

20. 5.4 The Binomial Theorem • Pascal’s Formula: orn-1C r-1 + n-1Cr = nCr • The Binomial Theorem:

21. Key Concepts • The coefficients of the terms when expanding (a+b)n match the number of terms in row n of Pascal’s Triangle Ex: 1 + 10x2 + 40x4 + 80x6 + 80x8 + 32x10 = (1+2x2)5 There are 6 coefficients here and since n=5, in row 5 of Pascal’s triangle there will be 6 terms • The number of terms in the expansion of (a+b)n is n+1

22. Examples • Rewrite using Pascal’s Formula: 12C8= ? Answer: 12C8 = 11C7 + 11C8 • Rewrite using Pascal’s Formula: 19C5 + 19C6 = ? Answer: 19C5 + 19C6 = 20C6 *Use n-1C r-1 + n-1Cr = nCr

23. Example • Expand: (a+b)6 = 6C0a6b0 + 6C1a5b1 + 6C2a4b2 +6C3a3b3 + 6C4a2b4 + 6C5a1b5 + 6C5a0b6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6 • Rewrite in the form of (a+b)n y12 + 8y9 + 24y6 + 32y3 + 16 7 terms => n=6 First term = y 12 => (y3)4 Last term = 16 => 24 Therefore in (a+b)n form the answer is: (y3 + 2)4

24. GAME TIME ;D

25. 5.1 • A recent survey of a group of students found that many participate in baseball, football, and soccer. The Venn diagram below shows the results of the survey. How many do not play soccer?

26. 5.2 SOLVE C 55 36

27. 5.3 • The game of euchre uses only 9s, 10s, jacks, Queens, kings, and aces from a standard deck of cards. How many five card hands have • All reds? • At least two red cards • At most red cards?

28. 5.4 Given (x+y)5, determine the value of k in the term: 10xky2

29. 5.1 • The Canadian Embassy in the Netherlands has 32 employees, all whom speak both French and English. In addition, 22 of the employees speak German and 15 speak Dutch and 10 speak both. How many of the employees speak neither German nor Dutch?

30. 5.2 • Change back into nCr form: 38! (38-7)!7!

31. 5.3 • A business is trying to decide on new business colours for their brochure. The employees can choose from peach, blue, green, orange, black, silver, gold, and red. They know that another business already has peach, blue, green, and orange. How many different combinations of four colours can the employees choose?

32. 5.4 When (a+x)6 is expanded, the first three coefficients of the terms are “1 + 3 + 3.75”. Determine the values of aandx.

33. 5.1 100 people • 75 read newspapers • 35 read books • 45 read magazines • 25 read both newspapers and books • 15 read both books and magazines • 10 read all three How many read both newspapers and magazines?

34. 5.2 • There are 4 female gym teachers and 7 male. They need 3 coaches for the school soccer team. In how many ways could they pick the coaches if there must be AT LEAST 1 female teacher?

35. 5.3 • In how many ways can a committee with six members form a subcommittee with at least one person in it?

36. 5.4 Using the Binomial Theorem, expand: (6x-5y)4

37. 5.4 Rewrite the following using Pascal’s formula: 18C9 – 17C9

38. 5.1

39. 5.2 • At a highschool reunion 12 friends met at the front gate. Everyone must greet and hug each other. If there are 12 friends, how many greetings take place?

40. 5.3 Determine whether the following questions involve permutations or combinations. • How many committees of 3 students can be formed from 12 students? • In how many ways can 12 skaters finish first, second, and third in a race? • How many outfits can you assemble from three pairs of pants, four shirts, and two pairs of shoes? • How many two-letter arrangements can be formed from the word star?

41. 5.4 What is the coefficient of a6b5, when (a + b)11 is expanded?

42. 5.1 Alex works at a radio station. He chooses some of the songs he will play based on the requests he receives. Alex made a list of 200 possible songs to play • All the songs in the top 100 • 125 rap songs • 60 phone in requests • 40 rap songs in the top 100 • 25 phone in requests in the top 100 • 30 phone in request for rap songs • Use a Venn diagram to determine How many phone in request were for rap songs in the top 100

43. 5.2 • At a concert, the singers must perform 50 songs in total. They must sing at least 8 songs within the first hour. They want to try and sing 15 so that the concert would finish earlier and she could go to the after party. In how many ways can they perform the song if the least they can perform is 8 and the most is 15?

44. 5.3 How many subsets are there for a set with • 2 distinct elements? • 4 distinct elements? • 7 distinct elements?

45. 5.4 Expand and simplify (x + 2/x2)4to determine the LAST TWO terms

46. All Done ;p Don’t forget to do review  Pg 296: # 1-14, 16bc, 17b, 23