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second/semester1/logseq-stuff/pages/Stars & Bars.md

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+- #[[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)
+-
+- 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
+	  card-last-interval:: 11.2
+	  card-repeats:: 3
+	  card-ease-factor:: 2.8
+	  card-next-schedule:: 2022-11-26T00:11:32.886Z
+	  card-last-reviewed:: 2022-11-14T20:11:32.887Z
+	  card-last-score:: 5
+		- 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**.
+		-
+	- **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.
+		- $$5^4 = 625$$
+-
+-