Page 116 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
2. Solving Problems Using Counting Principles
getting an odd number of heads from tossing
a coin 5 times are mutually exclusive), and
solve related probability problems [e.g., cal-
culate P(~A), P(A and B), P(A or B)] using a
2.2 solve simple problems using techniques for counting permutations and combinations, where all objects are distinct, and express the solutions using standard combinatorial
variety of strategies (e.g., Venn diagrams, n notation [e.g., n!, P(n, r), (r )] lists, formulas)
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
1.6 determine whether two events are indepen- dent or dependent and whether one event is conditional on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)
By the end of this course, students will:
2.1 recognize the use of permutations and combi- nations as counting techniques with advan- tages over other counting techniques (e.g., making a list; using a tree diagram; making a chart; drawing a Venn diagram), distinguish between situations that involve the use of per- mutations and those that involve the use of combinations (e.g., by considering whether or not order matters), and make connections between, and calculate, permutations and combinations
Sample problem: An organization with
10 members is considering two leadership models. One involves a steering committee with 4 members of equal standing. The other is an executive committee consisting of a president, vice-president, secretary, and treasurer. Determine the number of ways of selecting the executive committee from the 10 members and, using this number, the number of ways of selecting the steering committee from the 10 members. How are the calculations related? Use the calculations to explain the relationship between permuta- tions and combinations.
Sample problem: In many Aboriginal com- munities, it is common practice for people to shake hands when they gather. Use combina- tions to determine the total number of hand- shakes when 7 people gather, and verify using a different strategy.
2.3 solve introductory counting problems involv- ing the additive counting principle (e.g., determining the number of ways of selecting 2 boys or 2 girls from a group of 4 boys and
5 girls) and the multiplicative counting princi- ple (e.g., determining the number of ways of selecting 2 boys and 2 girls from a group of
4 boys and 5 girls)
2.4 make connections, through investigation, between combinations (i.e., n choose r) and
Pascal’s triangle [e.g., between (2)and rn
row 3 of Pascal’s triangle, between (2)and diagonal 3 of Pascal’s triangle]
Sample problem: A school is 5 blocks west and 3 blocks south of a student’s home. Determine, in a variety of ways (e.g., by drawing the routes, by using Pascal’s triangle, by using combinations), how many different routes the student can take from home to the school by going west or south at each corner.
2.5 solve probability problems using counting principles for situations involving equally likely outcomes
Sample problem: Two marbles are drawn randomly from a bag containing 12 green marbles and 16 red marbles. What is the probability that the two marbles are both green if the first marble is replaced? If the first marble is not replaced?
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