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Combinatorics: Combination without Repetition

Combinatorics: Combination without Repetition
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2018-12-06 18:30
Probability & StatisticsGeometric Construction
This example demonstrates calculating the number of combinations that can be formed by selecting rr objects from n different objects. It involves core combinatorics concepts like the combination formula nCr. GeoGebra commands used include nCr, Element, If, Sequence, Join, Polygon, with objects like Combine series and nCr. Suitable for high school mathematics teaching and combinatorics learning.

GeoGebra Commands & Parameters

  1. nCr(NObj, rObj) → nCr
  2. Element(AdjustSet, NObj) → Adjust
  3. If(nCr > CutValue, CutValue, nCr) → nlist1
  4. If(nCr > CutValue, nCr - CutValue, 0) → nlist2
  5. Element(SmallRSet, NObj) → SmallR
  6. Sequence(((a * cos(((360 * tt) * ° / nlist1))), (b * sin(((360 * tt) * ° / nlist1)))), tt, 1, nlist1) → OrbitLarge1
  7. Sequence(((a2 * cos(((360 * tt) * ° / nlist2))), (b2 * sin(((360 * tt) * ° / nlist2)))), tt, 1, nlist2) → OrbitLarge2
  8. Sequence(Sequence[(x(Element[OrbitLarge1, ss]) - Adjust + (SmallR * cos(((360 * tt) * ° / rr))), y(Element[OrbitLarge1, ss]) + (SmallR * sin(((360 * tt) * ° / rr)))), tt, 1, rr], ss, 1, nlist1) → OrbitSmall1
  9. Sequence(Sequence[(x(Element[OrbitLarge2, ss]) - Adjust + (SmallR * cos(((360 * tt) * ° / rr))), y(Element[OrbitLarge2, ss]) + (SmallR * sin(((360 * tt) * ° / rr)))), tt, 1, rr], ss, 1, nlist2) → OrbitSmall2
  10. Join({OrbitLarge1, OrbitLarge2}) → OrbitJoinLarge
  11. Join({OrbitSmall1, OrbitSmall2}) → OrbitJoinSmall
  12. Sequence((i - NObj / 2 - 0.8, 0), i, 1, NObj) → InitialPoint
  13. Sequence(Sequence[(i - NObj / 2 - 0.8, 0), i, 1, NObj], j, 1, nCr) → InitialPointSet
  14. Element(Element[Element[Element[Combine00, 6], 3], 2], 3) → test01
  15. Element(FinalPoint, 4) → test03
  16. Sequence(Sequence[(FindCor(tt, 0, 20, x(Element[Element[InitialPointSet, i], j]), x(Element[Element[FinalPointSet, i], j])), FindCor(tt, 0, 20, y(Element[Element[InitialPointSet, i], j]), y(Element[Element[FinalPointSet, i], j]))), j, 1, NObj], i, 1, nCr) → MovingPointSet
  17. Sequence(Sequence[(If[tt < i - 1, x(Element[Element[InitialPointSet, i], j]), If[tt > i, x(Element[Element[FinalPointSet, i], j]), FindCor(tt, i - 1, i, x(Element[Element[InitialPointSet, i], j]), x(Element[Element[FinalPointSet, i], j]))]], If[tt < i - 1, y(Element[Element[InitialPointSet, i], j]), If[tt > i, y(Element[Element[FinalPointSet, i], j]), FindCor(tt, i - 1, i, y(Element[Element[InitialPointSet, i], j]), y(Element[Element[FinalPointSet, i], j]))]]), j, 1, NObj], i, 1, nCr) → MovingPointSet2
  18. Element(Element[MovingPointSet, 2], 4) → test02
  19. Polygon(Point01, Point02, Point03, Point04) → poly1, point01, point02, point03, point04

AI Drawing Prompt Reference

Create a visual representation of combinatorial mathematics, featuring two concentric circular orbits of different sizes representing combinations of selecting r objects from n distinct ones. The larger orbit is divided into two parts, corresponding to the upper and lower limits of the combination number. Smaller orbits rotate around the larger ones, illustrating the distribution of combinations. At the center, there are two sets of points showing initial and final positions, connected by lines indicating movement paths. The overall design is suitable for teaching combinatorial formulas and probability geometry.
The prompt above was compiled from the original drawing commands and is for reference only.