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.

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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

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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

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Assets & Dependencies

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

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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!