Editing Hardy–Weinberg principle (section)

=== Generalization for more than two alleles ===
[[File:Hardy–Weinberg law - Punnett square2.svg|thumb|Punnett square for three-allele case (left) and four-allele case (right). White areas are homozygotes. Colored areas are heterozygotes.]]
Consider an extra allele frequency,&nbsp;''r''.  The two-allele case is the [[binomial theorem|binomial expansion]] of (''p''&nbsp;+&nbsp;''q'')<sup>2</sup>, and thus the three-allele case is the trinomial expansion of (''p''&nbsp;+&nbsp;''q''&nbsp;+&nbsp;''r'')<sup>2</sup>.

:<math>(p+q+r)^2=p^2 + q^2 + r^2 + 2pq +2pr + 2qr\,</math>

More generally, consider the alleles A<sub>1</sub>, ..., A<sub>''n''</sub> given by the allele frequencies ''p''<sub>1</sub> to ''p''<sub>''n''</sub>;

:<math>(p_1 + \cdots  + p_n)^2\,</math>

giving for all [[Zygosity|homozygotes]]:

:<math>f(A_i A_i) = p_i^2\,</math>

and for all [[Zygosity|heterozygotes]]:

:<math>f(A_i A_j) = 2p_ip_j\,</math>