Combinations Explained

Combinations

  • A combination is the number of arrangements possible when order does not matter.

    • Example: Choosing outfits where the order of putting on clothes (shirt before pants or vice versa) doesn't change the outfit.

  • The term "combination lock" is actually a misnomer since the order of numbers matters; it should be called a permutation lock.

Formula and Nomenclature

  • The mathematical notation for combinations is nCrnCr , where:

    • nn = the number of distinct objects.
    • rr = the number of nn objects taken.

  • The formula for calculating combinations is: nCr=n!r!(n−r)!nCr = \frac{n!}{r!(n-r)!}, where "!" denotes the factorial.

Example Problem: Vegetable Medley

  • Problem: Mister Rogers wants to make a vegetable medley for his family reunion using beets, carrots, tofu, daikon, and onions but can only use three ingredients.

  • Identify nn and rr:

    • n=5n = 5 (five distinct vegetables available).
    • r=3r = 3 (choosing three vegetables for the medley).

  • Apply the formula: 5C3=5!3!(5−3)!5C3 = \frac{5!}{3!(5-3)!}

Step-by-step Calculation

  1. Expand the factorials:
    • 5!=5×4×3×2×15! = 5 \times 4 \times 3 \times 2 \times 1
    • 3!=3×2×13! = 3 \times 2 \times 1
    • (5−3)!=2!=2×1(5-3)! = 2! = 2 \times 1
  2. Substitute into the formula: 5C3=5×4×3×2×1(3×2×1)(2×1)5C3 = \frac{5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(2 \times 1)}
  3. Simplify:
    • Cancel out common factors (e.g., 3×23 \times 2):
    • 5C3=5×425C3 = \frac{5 \times 4}{2}
  4. Calculate: 5C3=202=105C3 = \frac{20}{2} = 10
  • Result: There are 10 different combinations of vegetable medleys Mister Rogers can make using three out of the five available ingredients.

Listing Combinations

  • Examples of combinations:
    • Beet, tofu…
    • Carrot, tofu, daikon…
    • Tofu, daikon, and onion…
  • Listing all combinations would result in 10 different varieties.
  • Example of a combination that might be chosen: beet, daikon, and carrots.