r-Permutations P(n,r)

Combinatorics · Axiom Academy

Arranging r objects selected from n objects in order The key difference from regular permutations: We have n total objects available We select and arrange only r of them Order matters: ABC is different from BAC No repetition: each object can be used at most once Let's use the multiplication principle to derive P(n,r) by thinking about filling r positions: Position 2: We have (n-1) choices (one already used) Position 3: We have (n-2) choices Position r: We have (n-r+1) choices By the multiplication principle, we multiply all choices together. 3. Example: Selecting 3 from 5 Suppose we have 5 objects A, B, C, D, E and want to arrange 3 of them. Using the multiplication principle: Total: 5 × 4 × 3 = 60 arrangements What happens when we select all n objects (r = n)? This makes sense: arranging all n objects is just a regular permutation, which we already know is n!

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