#combinatorics — Public Fediverse posts

The lattice of sets of natural numbers is rich

https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/

Comments: https://news.ycombinator.com/item?id=49243687

#HackerNews #lattice #theory #natural #numbers #mathematics #rich #sets #combinatorics

The lattice of sets of natural numbers is rich

https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/

Comments: https://news.ycombinator.com/item?id=49243687

#HackerNews #lattice #theory #natural #numbers #mathematics #rich #sets #combinatorics

The lattice of sets of natural numbers is rich

https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/

Comments: https://news.ycombinator.com/item?id=49243687

#HackerNews #lattice #theory #natural #numbers #mathematics #rich #sets #combinatorics

The lattice of sets of natural numbers is rich

https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/

Comments: https://news.ycombinator.com/item?id=49243687

#HackerNews #lattice #theory #natural #numbers #mathematics #rich #sets #combinatorics

The lattice of sets of natural numbers is rich

https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/

Comments: https://news.ycombinator.com/item?id=49243687

#HackerNews #lattice #theory #natural #numbers #mathematics #rich #sets #combinatorics

Alright, future engineers!
**Combinations:** Choosing items where order *doesn't* matter.
Ex: Selecting 3 friends from 5 for a team. C(5,3) = 10 ways.
Pro-Tip: Think groups or sets. Changing the order doesn't create a new combination!
#Combinatorics #ProbStats #STEM #StudyNotes

Alright, future engineers!
**Permutations:** Selecting items from a set where *order matters*.
Ex: Arranging 3 items (ABC, ACB...). P(n,k) = n!/(n-k)!
Pro-Tip: Think specific sequences or rankings. Order defines a *new* permutation!
#ProbStats #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Permutations:** The number of ways to arrange a set of items where *order matters*.
Ex: Arranging 3 books (A,B,C) two at a time: AB, BA, AC, CA, BC, CB. (6 ways). `P(n,k) = n!/(n-k)!`
Pro-Tip: Order matters! is your key phrase to spot them!
#Combinatorics #ProbStats #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to choose 'r' items from 'n' where order *doesn't* matter.
Ex: Choosing 2 from 5 for a team: C(5,2) = 10. (C(n,r) = n!/(r!(n-r)!))
Pro-Tip: Think C for CHOOSE! Order is irrelevant. Essential for forming groups & probability.
#Combinatorics #Counting #STEM #StudyNotes

Alright, future engineers!
**Permutations:** Ways to arrange 'r' items from 'n' where order *matters*.
Ex: Arranging 3 books from 5 on a shelf: P(5,3) = 5! / (5-3)! = 60.
Pro-Tip: Think P for POSITION (order matters)! Crucial for sequencing tasks.
#Combinatorics #Probability #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to choose 'r' items from 'n' where order *doesn't* matter.
Ex: Choosing 3 members from 5 for a committee: C(5,3) = 5! / (3!2!) = 10.
Pro-Tip: Think C for CHOOSE (order doesn't matter)! Key for probability problems.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Permutations:** Ways to arrange items where order matters.
Ex: Arranging 3 people from 5 in seats: `P(5,3) = 5! / (5-3)! = 60`.
Pro-Tip: P for Position (order matters)! Crucial for scheduling & password analysis.
#Combinatorics #Probability #STEM #StudyNotes

Hey engineers!

**Pigeonhole Principle:** If you have more items than containers, at least one container *must* have >1 item.
Ex: 7 shirts in 6 drawers -> one drawer has >=2 shirts.
Pro-Tip: Simple, yet powerful for proofs and existence problems in #DiscreteMath!
#Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Permutation:** Ways to arrange 'r' items from 'n' where order matters.
Ex: P(n,r) = n!/(n-r)!. Arranging 3 people from 5: P(5,3)=60.
Pro-Tip: If ORDER matters (like passwords), it's a permutation!

#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Permutation:** An arrangement of items where the ORDER MATTERS.
Ex: Arrange 3 books: 3! = 6 ways.
Pro-Tip: Think ordered list. If order changes the outcome, it's a permutation!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combination:** Selecting items from a set where the *order of selection does not matter*.
Ex: Choosing 3 friends from 5 for a project. C(5,3) = 10 ways.
Pro-Tip: If reordering gives the same outcome, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Posting this a bit late, but I was very glad (and relieved) to hear that one of my former PhD students, Stijn Cambie, was granted one of this year's FWO Senior Postdoctoral Fellowships, to independently carry out his research programme for the next three years. (These are getting much harder to get, especially for fundamental, i.e. non-applied, research projects.)

https://www.fwo.be/en/results-outreach/announcement-of-results/275-new-postdoctoral-fellowships-awarded/

Congratulations Stijn!

#combinatorics #stijn #fwo #grants

Alright, future engineers!
**Permutation:** An arrangement of items where the *order of selection matters*.
Ex: Arranging 3 distinct books on a shelf from a set of 5. P(5,3) = 5!/(5-3)! = 60.
Pro-Tip: If reordering changes the outcome, it's a permutation!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Just dropped two new papers into the arxiv:

https://arxiv.org/abs/2607.12461

http://arxiv.org/abs/2607.17421

This caps off an extraordinary arc, the true seed of which was harshly interrupted by Covid, curling over multiple personal life changes, ending during a dawn in the dominance of AI-augmented mathematics.

Aside from the significant progress on one of my personal favourites ---indeed a *classic* Erdos problem--- what might be of wider interest is our use of (semiautomatically generated) Lean to corroborate the fidelity of computationally-aided proof methods (in this case, a nontrivial variant of flag algebras).

Oh, and let me extend my congratulations to all the Eoins.

#eoins #combinatorics #generativeAI #formalization #semiautomatic #lean #covid

Just dropped two new papers into the arxiv:

https://arxiv.org/abs/2607.12461

http://arxiv.org/abs/2607.17421

This caps off an extraordinary arc, the true seed of which was harshly interrupted by Covid, curling over multiple personal life changes, ending during a dawn in the dominance of AI-augmented mathematics.

Aside from the significant progress on one of my personal favourites ---indeed a *classic* Erdos problem--- what might be of wider interest is our use of (semiautomatically generated) Lean to corroborate the fidelity of computationally-aided proof methods (in this case, a nontrivial variant of flag algebras).

Oh, and let me extend my congratulations to all the Eoins.

#eoins #combinatorics #generativeAI #formalization #semiautomatic #lean #covid

Just dropped two new papers into the arxiv:

https://arxiv.org/abs/2607.12461

http://arxiv.org/abs/2607.17421

This caps off an extraordinary arc, the true seed of which was harshly interrupted by Covid, curling over multiple personal life changes, ending during a dawn in the dominance of AI-augmented mathematics.

Aside from the significant progress on one of my personal favourites ---indeed a *classic* Erdos problem--- what might be of wider interest is our use of (semiautomatically generated) Lean to corroborate the fidelity of computationally-aided proof methods (in this case, a nontrivial variant of flag algebras).

Oh, and let me extend my congratulations to all the Eoins.

#eoins #combinatorics #generativeAI #formalization #semiautomatic #lean #covid

Just dropped two new papers into the arxiv:

https://arxiv.org/abs/2607.12461

http://arxiv.org/abs/2607.17421

This caps off an extraordinary arc, the true seed of which was harshly interrupted by Covid, curling over multiple personal life changes, ending during a dawn in the dominance of AI-augmented mathematics.

Aside from the significant progress on one of my personal favourites ---indeed a *classic* Erdos problem--- what might be of wider interest is our use of (semiautomatically generated) Lean to corroborate the fidelity of computationally-aided proof methods (in this case, a nontrivial variant of flag algebras).

Oh, and let me extend my congratulations to all the Eoins.

#eoins #combinatorics #generativeAI #formalization #semiautomatic #lean #covid

Just dropped two new papers into the arxiv:

https://arxiv.org/abs/2607.12461

http://arxiv.org/abs/2607.17421

This caps off an extraordinary arc, the true seed of which was harshly interrupted by Covid, curling over multiple personal life changes, ending during a dawn in the dominance of AI-augmented mathematics.

Aside from the significant progress on one of my personal favourites ---indeed a *classic* Erdos problem--- what might be of wider interest is our use of (semiautomatically generated) Lean to corroborate the fidelity of computationally-aided proof methods (in this case, a nontrivial variant of flag algebras).

Oh, and let me extend my congratulations to all the Eoins.

#eoins #combinatorics #generativeAI #formalization #semiautomatic #lean #covid

Alright, future engineers!
**Combinations:** Ways to pick 'k' items from 'n' *where order doesn't matter*.
Ex: Choosing 3 people from 5 for a committee: C(5,3) = 5!/(3!2!) = 10 ways.
Pro-Tip: If order mattered, it'd be permutations! Key for probability.
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Selecting items where the order doesn't matter.
Ex: Choosing 2 team members from 5 people: 5C2 = 10 ways.
Pro-Tip: Look for choose or select. If swapping order doesn't make a new group, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combination:** An *unordered* selection of 'k' items from 'n' distinct items.
Ex: Choosing 3 people from 5 for a committee: `5C3 = 10` ways.
Pro-Tip: The *order* doesn't change the outcome! Think teams, not lineups.
#Combinatorics #Counting #STEM #StudyNotes

**Inclusion-Exclusion Principle:** Counts elements in a union of sets by adding individual sizes, then subtracting overlaps.
Ex: `|A U B| = |A| + |B| - |A n B|`
Pro-Tip: Essential for complex counting problems, prevents double-counting!

#Combinatorics #SetTheory #STEM #StudyNotes

Alright, future engineers!
**Permutations:** The number of ways to arrange items where ORDER MATTERS.
Ex: Arranging 3 books (A,B,C) on a shelf: 3! = 6 unique ways.
Pro-Tip: Think 'P' for Position or Podium! Order *always* counts.
#Combinatorics #DiscreteMath #STEM #StudyNotes

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

**Permutations:** The number of distinct ways to arrange a set of items where the order of arrangement matters.
Ex: Arranging 3 unique books on a shelf from a set of 5: `5P3 = 5!/(5-3)! = 60`.
Pro-Tip: Remember P for Position – order is key for permutations!
#Combinatorics #Probability #STEM #StudyNotes

Sum-product conjecture for real numbers is false: https://arxiv.org/abs/2605.28781

Given a finite set A ⊆ ℝ, the question was whether either the sumset A+A or the product set AA must be almost quadratic in size, i.e. max(|A+A|, |AA|) ≈ |A|² up to lower order terms. The conjecture said you can’t have strong additive and multiplicative structure at the same time

/1

#math #combinatorics

**Combinations:** Ways to choose items from a set where ORDER *doesn't* matter.
Ex: Picking 2 from {A,B,C} is {A,B}, {A,C}, {B,C} (3 ways). Formula: C(n,k)=n!/(k!(n-k)!)
Pro-Tip: Think groups or subsets – the sequence you pick them in doesn't change the group!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!

**Permutations:** Ways to arrange items from a set where ORDER MATTERS.
Ex: Arranging 3 distinct books on a shelf: 3! = 6 ways.
Pro-Tip: Use when position or sequence is crucial! Think passwords or schedules.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Permutation (nPr):** An arrangement of items where the order *matters*.
Ex: Arranging 3 distinct books from 5 on a shelf: P(5,3) = 60 ways.
Pro-Tip: Think 'P' for 'Position' – a different order means a different outcome!
#Probability #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combination:** A selection of items where the order of selection *doesn't matter*.
Ex: Choosing 3 people from 10 for a committee: C(10,3) = 120.
Pro-Tip: Think 'C' for 'Committee' – selecting a group where roles aren't specified.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combination:** Selecting items where order *doesn't* matter.
Ex: Choosing 3 toppings from 10. C(10,3) = 120.
Pro-Tip: 'C' for 'Choice' – order isn't relevant to the selection itself!
#Combinatorics #DiscreteMath #STEM #StudyNotes

The past two months, I helped coordinate the "Phase Transitions..." research semester programme at CWI (https://www.cwi.nl/en/events/research-semester-programmes/phasecap-phase-transitions-in-combinatorics-algorithms-probability/). It ended last Friday, and still I feel "hungover" from the intensive blur of activities/developments/ideas. Very grateful to my team --Feri, Jop, Serte, Carla, Noela, Guus-- we did it!

In parallel, during the same two months, after the dawn of recognition of what has arrived (after a tip from Jeroen), I underwent a kind of phase transition myself. Avowed refusenik in March (see https://mathstodon.xyz/@kangmeister/115252549665766971); an "anti-Gemini" research working group in April; compulsive button-pressing in May. (And yes, I *know* it is easy to set it up for pressing fewer buttons...)

That poetic part of me (or whatever remains of it) is allured by the term, "cognitive surrender", if only to help in my search for the right words to describe the sharp changes underway in various facets of mathematical life/growth.

#CWI #combinatorics #algorithms #probability #conferences #generativeAI #formalization #lean #scientificpublishing

The past two months, I helped coordinate the "Phase Transitions..." research semester programme at CWI (https://www.cwi.nl/en/events/research-semester-programmes/phasecap-phase-transitions-in-combinatorics-algorithms-probability/). It ended last Friday, and still I feel "hungover" from the intensive blur of activities/developments/ideas. Very grateful to my team --Feri, Jop, Serte, Carla, Noela, Guus-- we did it!

In parallel, during the same two months, after the dawn of recognition of what has arrived (after a tip from Jeroen), I underwent a kind of phase transition myself. Avowed refusenik in March (see https://mathstodon.xyz/@kangmeister/115252549665766971); an "anti-Gemini" research working group in April; compulsive button-pressing in May. (And yes, I *know* it is easy to set it up for pressing fewer buttons...)

That poetic part of me (or whatever remains of it) is allured by the term, "cognitive surrender", if only to help in my search for the right words to describe the sharp changes underway in various facets of mathematical life/growth.

#CWI #combinatorics #algorithms #probability #conferences #generativeAI #formalization #lean #scientificpublishing

The past two months, I helped coordinate the "Phase Transitions..." research semester programme at CWI (https://www.cwi.nl/en/events/research-semester-programmes/phasecap-phase-transitions-in-combinatorics-algorithms-probability/). It ended last Friday, and still I feel "hungover" from the intensive blur of activities/developments/ideas. Very grateful to my team --Feri, Jop, Serte, Carla, Noela, Guus-- we did it!

In parallel, during the same two months, after the dawn of recognition of what has arrived (after a tip from Jeroen), I underwent a kind of phase transition myself. Avowed refusenik in March (see https://mathstodon.xyz/@kangmeister/115252549665766971); an "anti-Gemini" research working group in April; compulsive button-pressing in May. (And yes, I *know* it is easy to set it up for pressing fewer buttons...)

That poetic part of me (or whatever remains of it) is allured by the term, "cognitive surrender", if only to help in my search for the right words to describe the sharp changes underway in various facets of mathematical life/growth.

#CWI #combinatorics #algorithms #probability #conferences #generativeAI #formalization #lean #scientificpublishing

The past two months, I helped coordinate the "Phase Transitions..." research semester programme at CWI (https://www.cwi.nl/en/events/research-semester-programmes/phasecap-phase-transitions-in-combinatorics-algorithms-probability/). It ended last Friday, and still I feel "hungover" from the intensive blur of activities/developments/ideas. Very grateful to my team --Feri, Jop, Serte, Carla, Noela, Guus-- we did it!

In parallel, during the same two months, after the dawn of recognition of what has arrived (after a tip from Jeroen), I underwent a kind of phase transition myself. Avowed refusenik in March (see https://mathstodon.xyz/@kangmeister/115252549665766971); an "anti-Gemini" research working group in April; compulsive button-pressing in May. (And yes, I *know* it is easy to set it up for pressing fewer buttons...)

That poetic part of me (or whatever remains of it) is allured by the term, "cognitive surrender", if only to help in my search for the right words to describe the sharp changes underway in various facets of mathematical life/growth.

#CWI #combinatorics #algorithms #probability #conferences #generativeAI #formalization #lean #scientificpublishing

Alright, future engineers!
**Permutations:** Ways to arrange items where order *matters*.
Ex: Arranging 3 distinct books on a shelf: 3! = 6 ways.
Pro-Tip: Think 'P' for 'Position' – each arrangement is unique!
#StatsProb #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to select items where order *doesn't* matter.
Formula: C(n, k) = n! / (k! * (n-k)!)
Pro-Tip: Think 'C' for 'Committee' - the order you pick members doesn't change the committee!
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Permutations:** Ways to arrange items where order *DOES* matter.
Ex: Arranging 3 books from 5 on a shelf: P(5,3) = 60 ways.
Pro-Tip: Think 'P' for 'Position'! Order matters for unique arrangements.
#StatsProb #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations:** Ways to choose items where ORDER does *NOT* matter.
Ex: Choosing 3 books from 5 for a read list: C(5,3) = 10 ways.
Pro-Tip: Think 'C' for 'Choice'! Grouping where order is irrelevant.
#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Combinations:** The number of ways to choose items from a set where the ORDER DOES NOT matter.
Ex: Picking 3 teammates from 5 for a project: `C(5,3) = 10` ways.
Pro-Tip: Think 'C' for 'Committee'! If swapping selected items doesn't change the group, it's a Combination.
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Permutations:** The number of ways to arrange items where the ORDER matters.
Ex: How many ways to pick & arrange 3 out of 5 engineers for 3 distinct roles? P(5,3) = 60
Pro-Tip: Think P for Position! If swapping two items changes the outcome, it's a Permutation.
#Combinatorics #Probability #STEM #StudyNotes

Registration is open for the 2026 colloquia in #combinatorics on 13+14 of May, 2026, in London.
#scientificConference #mathematics
https://2dcic.github.io/

Registration is open for the 2026 colloquia in #combinatorics on 13+14 of May, 2026, in London.
#scientificConference #mathematics
https://2dcic.github.io/

Registration is open for the 2026 colloquia in #combinatorics on 13+14 of May, 2026, in London.
#scientificConference #mathematics
https://2dcic.github.io/

Registration is open for the 2026 colloquia in #combinatorics on 13+14 of May, 2026, in London.
#scientificConference #mathematics
https://2dcic.github.io/

Alright, future engineers!
**Combinations:** Ways to *choose* items from a set where the order *doesn't* matter.
Ex: Picking 3 teammates from 5 friends: `C(5,3) = 10` ways.
Pro-Tip: If swapping items doesn't create a new outcome, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations**: ways to choose items where order *doesn't* matter.
Ex: Choose 2 teammates from 5 friends: `C(5,2) = 5!/(2!3!) = 10` ways.
Pro-Tip: Think 'choosing a committee' – the group is what matters, not selection order!
#Probability #Combinatorics #STEM #StudyNotes

Alright, future engineers!

**Combinations**: ways to choose items where order *doesn't* matter.
Ex: Picking 3 committee members from 5 people: C(5,3) = 10 ways.
Pro-Tip: Think 'selecting ingredients for a soup' – the order you add them doesn't change the final soup!

#Combinatorics #DiscreteMath #STEM #StudyNotes

Alright, future engineers!

**Permutations**: ways to arrange items where order matters.
Ex: Arranging 3 books (A,B,C) is 3! = 6 ways.
Pro-Tip: Think 'President, VP, Secretary' - roles are distinct!

#Probability #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations** count selections where order *doesn't* matter. Ex: Picking 3 teammates from 10. Formula: C(n,r) = n! / (r!(n-r)!). Pro-Tip: If reordering items yields the *same* group, it's a combination!
#Probability #Combinatorics #STEM #StudyNotes

Alright, future engineers!
**Combinations** count selections where order *doesn't* matter. Ex: Picking 3 teammates from 10. Formula: C(n,r) = n! / (r!(n-r)!). Pro-Tip: If reordering items yields the *same* group, it's a combination!
#Probability #Combinatorics #STEM #StudyNotes