(2) We may also determine genotype frequencies involving only two of the several alleles in a multiple allelic series. As an example we may take 'A_{1}' and ' A_{2}' as two such alleles and their corresponding gene frequencies as q_{1}and q_{2 }respectively so that q_{1} + q_{2} ≠ 1, because frequencies of several other alleles are not included. The genotypic frequencies involving the two alleles at the equilibrium stage can be written in the form of Hardy Weinberg Equilibrium Equation as : q_{1}^{2} A_{1}A_{1 }+ 2q_{1}q_{2}A_{1}A_{2 }+ q_{2}^{2}A_{2}A_{2}. In this case, neither the total gene frequency of 'A_{1}' and 'A_{2}' will be unity nor will the frequencies of genotypes involving the two alleles be unity.
(3) In the above two conditions, the multiple allelicseries was reduced to a two allele series for purpose of simplification. However, it is possible to determine frequencies of genotypes involving three or more alleles at the equilibrium stage. In this case, each allelic frequency will have to be considered as an element in a multinomial expansion. For instance, if there are three alleles 'A_{1}', 'A_{2}' and 'A_{3}' at the locus, with their corresponding gene frequencies 'p', 'q' and 'r' respectively, then p+q+r = 1. In such a case the trinomial expansion i.e. (p+q+r)^{2} = p^{2} A_{1}A_{1} + 2pq A_{1}A_{2 }+ 2pr A_{1}A_{3} + q^{2} A_{2}A_{2} + 2qr A_{2}A_{3} + r^{2} A_{3}A_{3} will represent the genotypic frequencies at the equilibrium stage. It can be realized that in this case zygotic combinations between haploid gametes will depend upon frequencies of individual alleles because each haploid gamete contains a single allele at a particular locus as shown in Figure 46.3. Equilibrium will be established in this case also within a single generation of random mating. 


Fig. 46.3. Frequencies of genotypes generated under conditions of random mating when there are three alleles A_{1}, A_{2}, A_{3}present at a locus, and their gene frequencies are p = 0.2, q = 0.5 and r = 0.3. 
