Chapter 5 - Combinations and the Binomial Theorem



Chapter 5 - Combinations and the Binomial Theorem

Things You Should Know

1. Venn Diagrams

a) Create a series of circles to illustrate the different sets of data.

b) Determine the elements common to multiple sets first.

c) Subtract the common elements from the individual ones to determine the data in each.

d) Whatever is not included in the circles, belongs to the null set. (Belongs to none of the sets)

Ex/ Of the 140 grade 12 students at Churchill, 52 have signed up for biology, 71 for chemistry and 40 for physics. The science students include 15 who are taking both biology and chemistry, 8 who are taking chemistry and physics, 11 who are taking biology and physics, and 2 who are taking all three sciences.

2. Combinations

a)[pic]

b) Combinations apply to the situation where one wants a selection of items in no particular order.

c) If there are n distinct items, the total combinations containing at least 1 item from each group is[pic].

d) The total number of selections that can be made from p items of one kind, q items of another kind, r items of another kind, and so on is given by [pic]

3. The Binomial Theorem

a) [pic]

Ex/ Expand using the Binomial Theorem: [pic]

[pic]

b) There are [pic]terms in the expansion.

c) The exponents on the terms in the binomial expansion sum to n.

Chapter 7 - Practice

1. There are 150 books in a sale bin, 63 are fiction, 51 are Canadian, and 85 are about the military. Of these 22 books are Canadian fiction, 17 are about the Canadian military, 26 are military fiction, and 11 books fit in all three categories. How many books do not fit into any of these categories?

2. If you are going on a five-day trip, in how many ways can you pick five pairs of socks from a drawer in which there are eight pairs of socks?

3. Find the number of different 5-card hands that can be dealt from a deck of 52 cards.

4. The school gardening club consists of 5 boys and 5 girls. How many working groups of four people can be formed with

a) No restrictions

b) Four boys

c) Three boys and a girl

d) Two boys and two girls

e) At least 1 boy

5. From a group of 14 Conservatives, 12 Liberals, 8 NDP, and 2 Independent Members of Parliament, how many different committees can be formed consisting of three Conservatives, three Liberals, two NDP and one Independent?

6. George arrives at the giant auction sale late in the afternoon. There are only 5 items left to be sold. How many different purchases could he make?

7. How many different sums of money can be made from four $5 bills, three $10 bills, and five $20 bills?

8. Nick is responsible for keeping the office drinks cooler filled. He can purchase up to six cases of carbonated drinks, four cases of juices, five cases of spring water, and five cases of iced tea without sending the purchase order to his supervisor for approval. How many different purchases can Nick make without his supervisor’s signature?

9. You and your lab partner have made a list of six tasks you must do to complete your science project. In how many ways can you divide this work? Explain your reasoning.

10. Expand and simplify the following[pic].

11. Find the first 4 terms in the expansion of[pic].

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2

9

6

13

Chem

Bio

Phys

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