- #[[MA284 - Discrete Mathematics]]
- **Previous Topic:** [[Combinatorial Proofs]]
- **Next Topic:** [[Advanced PIE, Derangements, & Counting Functions]]
- **Relevant Slides:** ![MA284-Week05.pdf](../assets/MA284-Week05_1664971430860_0.pdf)
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- Example: How many ways can you give 7 apples to 4 lecturers?
	- How many ways can you arrange 3 bars out of 7 stars and 3 bars (10)?
	- $$* | * | * | * * *$$
	- $$\binom{10}{3} = 120$$
- # Multisets vs Sets
	- What is a **multiset**? #card
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		- A **multiset** is a set of objects, where each object can appear more than once.
		- As with an ordinary set, order doesn't matter.
	- **Set:** Neither order nor repetition of elements matters.
		- e.g., $\{a,b,c\} = \{c,a,b\} = \{c,c,a,b,a,b,c\}$
	- **Multiset:** Order does not matter, but we count the **multiplicity** (number of times it occurs) of each element.
		- e.g., $\{a,b,c\} \neq \{c,c,a,b,a,b,c\}$, provided they are **multisets**.
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	- **Example:** How many **multisets** of size 4 can you form using numbers $\{1,2,3,4,5\}$?
		- Let's answer this using stars & bars.
		- e.g.:
			- $\{1,2,3,4\} = * | ** | |*|$
			- $\{5,3,3,1\} = *||**||*$
			- Each multiset can be represented using 8 boxes & 4 stars.
		- $$5^4 = 625$$
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