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Permutations vs Combinations

対象: High Schoolカテゴリ: Math

説明

This animation contrasts permutations (order matters) with combinations (order doesn't matter) using a concrete example of choosing 2 items from {A, B, C}. A tree diagram visually enumerates all 6 permutations, then groups are collapsed to show only 3 combinations, making the divide-by-r! factor intuitive.

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Permutations vs Combinations

Description

This animation contrasts permutations (order matters) with combinations (order doesn't matter) using a concrete example of choosing 2 items from {A, B, C}. A tree diagram visually enumerates all 6 permutations, then groups are collapsed to show only 3 combinations, making the divide-by-r! factor intuitive.

Phases

# Phase Name Duration Description
1 Introduction 4s Title; pose the question: "Choose 2 from {A, B, C}"
2 Tree diagram 10s Build tree level by level: root → first pick (A/B/C) → second pick (branch outcomes)
3 Permutations 8s Highlight all 6 leaf nodes (AB, AC, BA, BC, CA, CB); show P(3,2) = 6 formula
4 Collapse to combinations 8s Draw brackets grouping {AB,BA}, {AC,CA}, {BC,CB}; fade duplicates; show 3 remain
5 Formula display 8s Show P and C formulas side by side; numeric substitution
6 Summary 6s Key insight: C = P / r! to avoid counting duplicates

Layout

+-----------------------------------------------+
|  Permutations vs Combinations     [title]     |
|                                                |
|         {A, B, C}                             |
|        /    |    \                            |
|       A     B     C      [first pick]         |
|      / \   / \   / \                          |
|     B   C A   C A   B    [second pick]        |
|    AB  AC BA  BC CA  CB  [leaf outcomes]       |
|                                                |
|  P(3,2) = 6   C(3,2) = 3   [formula row]     |
+-----------------------------------------------+

Area Descriptions

Area Content Notes
Top Title text Fades to corner after phase 1
Center Tree diagram nodes + edges Nodes are Text/Circle objects
Leaf row Outcome labels Colored GREEN for permutations, then some RED for duplicates
Bottom P and C formulas Two MathTex objects side-by-side

Assets & Dependencies

  • Fonts: LaTeX
  • Colors: PERM_COLOR = GREEN, COMBO_COLOR = BLUE, DUPLICATE_COLOR = RED, BG = #1a1a2e
  • Manim version: ManimCE 0.19.1

Notes

  • Tree is constructed manually with Line objects between node centers
  • Duplicate outcomes (BA when AB already counted) get a cross-out or fade
  • Formulas use MathTex for proper rendering of n!, (n-r)!, r!