33 lines
1.4 KiB
Markdown
33 lines
1.4 KiB
Markdown
- #[[MA284 - Discrete Mathematics]]
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- **Previous Topic:** [[Combinatorial Proofs]]
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- **Next Topic:** [[Advanced PIE, Derangements, & Counting Functions]]
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- **Relevant Slides:** 
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- Example: How many ways can you give 7 apples to 4 lecturers?
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- How many ways can you arrange 3 bars out of 7 stars and 3 bars (10)?
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- $$* | * | * | * * *$$
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- $$\binom{10}{3} = 120$$
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- # Multisets vs Sets
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- What is a **multiset**? #card
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card-last-interval:: 11.2
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card-repeats:: 3
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card-ease-factor:: 2.8
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card-next-schedule:: 2022-11-26T00:11:32.886Z
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card-last-reviewed:: 2022-11-14T20:11:32.887Z
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card-last-score:: 5
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- A **multiset** is a set of objects, where each object can appear more than once.
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- As with an ordinary set, order doesn't matter.
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- **Set:** Neither order nor repetition of elements matters.
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- e.g., $\{a,b,c\} = \{c,a,b\} = \{c,c,a,b,a,b,c\}$
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- **Multiset:** Order does not matter, but we count the **multiplicity** (number of times it occurs) of each element.
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- 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 $\{1,2,3,4,5\}$?
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- Let's answer this using stars & bars.
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- e.g.:
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- $\{1,2,3,4\} = * | ** | |*|$
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- $\{5,3,3,1\} = *||**||*$
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- Each multiset can be represented using 8 boxes & 4 stars.
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- $$5^4 = 625$$
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