#Combinatorics #Year11 #Ext1 >[!info]- [Permutations and combinations | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/n11/fa40471150) >- ME1-11-04 uses permutations and combinations to solve problems involving counting, ordering and probability ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | --------------------------------------------- | -------------------------------- | -------------------------------- | | [[Probability and Data\|Probability]] | - probability | - probability with combinatorics | ## Permutations and combinations - Use the notation $n!$ (read as n factorial), where $n!=n(n-1)(n-2)\times\ldots\times3\times2\times1$ 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 $^nP_r$ to represent an ordered selection of $r$ objects from $n$ distinct objects and observe that $^nP_n=n!$ and $^nP_0=1$ - Use the multiplication principle to establish that the number of ordered selections of $r$ objects from $n$ distinct objects is $n(n-1)(n-2)\times\ldots\times(n-r+1)$ and show that $^nP_r=n(n-1)(n-2)\times\ldots\times(n-r+1)=\frac{n!}{(n-r)!}$ - 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 $^nC_r$ or $\binom{n}{r}$ 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 $^nC_r=\frac{n!}{r!(n-r)!}$ - Show that $^nC_n=^nC_0=1$ and $^nC_1=^nC_{n-1}=n$ - Show that $^nC_r=^nC_{n-r}$, 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 $^nC_r=^{n-1}C_{r-1}+^{n-1}C_r$ 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