Generalization for more than two alleles Hardy–Weinberg principle

punnett square three-allele case (left) , four-allele case (right). white areas homozygotes. colored areas heterozygotes.

consider allele frequency, r. two-allele case binomial expansion of (p + q), , three-allele case trinomial expansion of (p + q+ r).

(
p
+
q
+
r

)

2


=

p

2


+

q

2


+

r

2


+
2
p
q
+
2
p
r
+
2
q
r



{\displaystyle (p+q+r)^{2}=p^{2}+q^{2}+r^{2}+2pq+2pr+2qr\,}

more generally, consider alleles a1, ..., given allele frequencies p1 pn;

(

p

1


+
⋯
+

p

n



)

2





{\displaystyle (p_{1}+\cdots +p_{n})^{2}\,}

giving homozygotes:

f
(

a

i



a

i


)
=

p

i


2





{\displaystyle f(a_{i}a_{i})=p_{i}^{2}\,}

and heterozygotes:

f
(

a

i



a

j


)
=
2

p

i



p

j





{\displaystyle f(a_{i}a_{j})=2p_{i}p_{j}\,}