Factorial, Permutations, Combinations (distinct objects; no repeats)

Factorial, Permutations & Combinations

1) n! (factorial)

  • Meaning: tells you how many permutations (complete orderings) you can generate with n objects at your disposal.
  • Definition: n! = n × (n−1) × (n−2) × … × 2 × 1, with 0! = 1.
  • Example: 5! = 5×4×3×2×1 = 120.

2) n!/(n−r)! (permutations of r choices from n; order matters)

  • Meaning: tells you how many ordered outcomes you can generate when you make r choices out of a collection of n objects, without reuse.
  • How to see it:
    1st choice: n options
    2nd choice: (n−1) options
    3rd choice: (n−2) options
    …
    rth choice: (n−r+1) options
    Multiply: n × (n−1) × … × (n−r+1) = n!/(n−r)!.
  • Example (n=5, r=2): 5P2 = 5!/(5−2)! = 5!/3! = (5×4×3×2×1)/(3×2×1) = 5×4 = 20.

3) (n!/(n−r)!)/r! = n!/((n−r)! r!) (combinations; order neglected)

  • Meaning: tells you how many selections you can make when choosing r objects from n, where order does not matter.
  • Key idea: First, count the r-choice permutations: n!/(n−r)!. But when order is neglected, many of those “different” outcomes are the same selection written in a different order. For any fixed selection of r objects, there are exactly r! permutations of those same r objects. Therefore, each selection has been counted r! times, so dividing by r! eliminates repeats.
  • Example (n=5, r=2): 5C2 = (5P2)/2! = 20/2 = 10.

One-line summary

  • Order matters (arrangements): nPr = n!/(n−r)!
  • Order doesn’t matter (selections): nCr = n!/((n−r)! r!)