Permutations and Combinations

combinatorics year11 ext1

Permutations and combinations | NSW Curriculum Website

• ME1-11-04 uses permutations and combinations to solve problems involving counting, ordering and probability

📖 Prior Knowledge

Content	Prior knowledge	Used for
Probability	- probability	- probability with combinatorics

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 $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ 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 $0\le r\le n$ 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 $1\le r\le n-1$ 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