1.4 KiB
1.4 KiB
- #MA284 - Discrete Mathematics
- Previous Topic: Combinatorial Proofs
- Next Topic: Advanced PIE, Derangements, & Counting Functions
- Relevant Slides:
- 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\}
- e.g.,
- 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.
- e.g.,
- 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
- 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
