Back to Algebra repository
Algebra2D

Combinatorics and Probability - Theory & Concepts

Fundamental counting principles, permutations, combinations, and basic probability theory.

Open the complete lesson

Permutations vs. Combinations Explorer

Pool of distinct items to choose from
Number of items to select
(Note: r cannot exceed n. Assumes distinct items, no duplicates)

Item Pool (4):

1
2
3
4

Permutations (Order Matters)

Scenario: Creating a password, or picking 1st/2nd/3rd place.
(Choosing 2 items where order matters)

P(4,2)=4!(4−2)!=242=12P(4, 2) = \frac{4!}{(4-2)!} = \frac{24}{2} = 12

12 distinct arrangements

1,21,31,42,12,32,43,13,23,44,14,24,3

Combinations (Order Does NOT Matter)

Scenario: Forming a committee, or drawing a hand of cards.
(Choosing 2items where order doesn't matter)

C(4,2)=4!2!(4−2)!=242⋅2=6C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{24}{2 \cdot 2} = 6

6 distinct groups

1,21,31,42,32,43,4

Notice that P(4,2)P(4, 2) is always exactly 2!2! times larger than C(4,2)C(4, 2), because for every combination (group), there are 2!2! ways to arrange those specific items. (2!=22! = 2).

Related simulations

Fundamentals - Theory & ConceptsFundamentals - Theory & Concepts - Absolute ValueExponents and Radicals - Theory & Concepts - Exponent RulesExponents and Radicals - Theory & Concepts - Power Function