Q.62 Hardy Weinberg’s equilibrium for a locus with two alleles p and q is mathematically defined as
P² + Q² + 2PQ = 1.
Which of the following equations represents the corresponding equilibrium for a locus with three alleles
p, q and r?
(P, Q and R represent the frequencies ofp, q and r, respectively)

1. P³ + Q³ + R³ + 3PQR = 1
2. P² + Q² + R² + 2PQ + 2QR + 2PR = 1
3. P²Q + Q²R + R²P + 2PQ + 2QR + 2PR = 1
4. P² + Q² + R² + 2P²Q + 2Q²R + 2P²R = 1

Hardy-Weinberg equilibrium describes stable allele and genotype frequencies in populations without evolutionary forces. For three alleles (P, Q, R), the equation follows the trinomial expansion (P + Q + R)² = 1.

Correct Answer

Option (B) P² + Q² + R² + 2PQ + 2QR + 2PR = 1 represents the equilibrium. This expands (P + Q + R)², where P + Q + R = 1 sums allele frequencies, matching the two-allele case (P + Q)² = P² + 2PQ + Q² = 1.

Option Analysis

• (A) P³ + Q³ + R³ + 3PQR = 1: Incorrect; this is (P + Q + R)³ expansion for haploid or cube terms, not diploid genotypes.
• (B) P² + Q² + R² + 2PQ + 2QR + 2PR = 1: Correct; homozygous frequencies (P², Q², R²) plus twice heterozygous pairs (2PQ, 2PR, 2QR) sum to 1.
• (C) P²Q + Q²R + R²P + 2PQ + 2QR + 2PR = 1: Incorrect; mixes cyclic heterozygote-like terms (P²Q) with standard ones, doesn’t match expansion.
• (D) P² + Q² + R² + 2P²Q + 2Q²R + 2P²R = 1: Incorrect; uses squared dominant-like terms (2P²Q), violating heterozygote equality.

Genotype Frequencies

Under equilibrium, frequencies are:

Homozygotes: P² (PP), Q² (QQ), R² (RR)

Heterozygotes: 2PQ (PQ), 2PR (PR), 2QR (QR)