11–12Mathematics Extension 1 11–12 Syllabus

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Year 11

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Permutations and combinations

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME1-11-04
uses permutations and combinations to solve problems involving counting, ordering and probability

Permutations and combinations

Use the notation $n!$ (read as $n$ factorial), where $n! = n (n - 1) (n - 2) \times \dots \times 3 \times 2 \times 1$ for positive integers $n$
Use $n! = n \times (n - 1)!$ and the convention $0! = 1$ in calculations and to simplify algebraic expressions involving factorials
Establish and use the multiplication principle: if a selection can be made in two stages, where there are $m$ choices for the first stage and $n$ choices for the second stage then there are $m \times n$ choices for the selection
Apply the multiplication principle to explain why the number of ways of ordering $n$ distinct objects in a straight line is $n!$
Define a permutation as an ordered selection of some or all objects from a set of distinct objects
Use the notation $^{n} P_{r}$ to represent an ordered selection of $r$ objects from $n$ distinct objects and observe that $^{n} P_{n} = n!$ and $^{n} P_{0} = 1$
Use the multiplication principle to establish that the number of ordered selections of $r$ objects from $n$ distinct objects is $n (n - 1) (n - 2) \times \dots \times (n - r + 1)$ and show that $^{n} P_{r} = n (n - 1) (n - 2) \times \dots \times (n - r + 1) = \frac{n!}{(n - r)!}$
Solve problems involving permutations, including situations where the objects are not all distinct
Solve problems involving permutations with restrictions on the placement of one or more objects
Explain why the number of ways to arrange $n$ distinct objects in a circle is $(n - 1)!$
Solve problems involving circular arrangements of distinct objects with or without restrictions on the placement of one or more objects
Define a combination and use the notation $^{n} C_{r}$ or $(\begin{matrix} n \\ r \end{matrix})$ to represent the number of ways of selecting a subset of $r$ objects from $n$ distinct objects, where order is not important
Establish and use the formula $^{n} C_{r} = \frac{n!}{r! (n - r)!}$
Show that $^{n} C_{n} =^{n} C_{0} = 1$ and $^{n} C_{1} =^{n} C_{n - 1} = n$
Show that $^{n} C_{r} =^{n} C_{n - r}$ , for $0 \leq r \leq n$ by selecting $r$ objects from $n$ distinct objects for inclusions and $n - r$ objects from $n$ distinct objects for exclusion
Prove $^{n} C_{r} =^{n - 1} C_{r - 1} +^{n - 1} C_{r}$ for $1 \leq r \leq n - 1$ algebraically and using combinatorial arguments
Solve problems involving combinations with or without restrictions on the selection of one or more objects
Solve problems involving both permutations and combinations, including problems which require consideration of cases
Solve probability problems involving permutations and combinations

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Year 12

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11–12Mathematics Extension 1 11–12 Syllabus

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