A derivation of Fermi and Bose statistics is given, based on the general structure of quantum mechanics, together with a simple axiom of direct physical significance. The axiom concerns an operation, denoted by ∘, forming the union of two states; ψ ∘φ denotes the state of a compound system whose parts are in the states ψ and φ. Let ψ and φ denote 1-particle states and assume: (a) ψ ∘φ exists whenever ψ ⊥ φ; (b) ψ ∘φ=φ ∘ψ; (c) the transition probability between ψ ∘φ and ψ′ ∘φ′ is zero if ψ ⊥ ψ′ and ψ ⊥ φ′, and (d) the product of the transition probabilities from ψ to ψ′ and from φ to φ′ if ψ ⊥ φ′ and φ ⊥ ψ′. It is then shown that, at least in so far as 2-particle states are concerned, the particles obey either Fermi or Bose statistics.