Solution

Human skin colour is an example of polygenic (multiple gene) inheritances. Assume that three "dominant" capital letter genes (A, B and C) control dark pigmentation because more melanin is produced. The "recessive"alleles of these three genes (a, b & c) control light pigmentation because lower amounts of melanin are produced. The words dominant and recessive are placed in quotation marks because these pairs of alleles are not truly dominant and recessive as in some of the garden pea traits that Gregor Mendel studied. A genotype with all "dominant" capital genes (AABBCC) has the maximum amount of melanin and very dark skin. A genotype with all "recessive" small case genes (aabbcc) has the lowest amount of melanin and very light skin. Each "dominant" capital gene produces one unit of color so that a wide range of intermediate skin colours are produced, depending on the number of "dominant" capital genes in the genotype. For example, a genotype with three "dominant" capital genes and three small case "recessive" genes (AaBbCc) has a medium amount of melanin and an intermediate skin color. This latter genotype would be characteristic of a mulatto.
In the following cross between two mulatto genotypes (AaBbCc x AaBbCc), each parent produces eight different types of gametes and these gametes combine with each other in 64 different ways resulting in a total of seven skin colours. The skin colours can be represented by the number of capital letters, ranging from zero (no capital letters) to six (all capital letters).
Note: Skin colour may involve at least four pairs of alleles with nine (or more) shades of skin colour.
The cross can be shown with the binomial expansion (a + b)6 where the letter a = number of capital letters and the letter b = number of small case letters. Each term in the expression represents the number of offspring with a specific skin colour phenotype based on the number of capital letters in the genotype. For example, 20 offspring have three capital letters in their genotype and have a skin colour that is intermediate between very dark with all caps (AABBCC) and very light with no caps (aabbcc).
(a + b)6 = a6 + 6 a5b + 15 a4b2 + 20 a3b3 + 15 a2b4 + 6 ab5 + b6
Multiple gene (polygenic) inheritance explains many plant and animal traits where there is a wide variation between extreme phenotypes, with most individuals having intermediate phenotypes. In polygenic inheritance the "dominant" capital genes are additive, each capital gene adding one unit of color to the genotype. With more capital genes, the phenotype (appearance) gets darker. The garden peas studied by Gregor Mendel involved pairs of alleles with only three possible genotypes and two phenotypes per trait. For example, the gene for round pea (R) is dominant over the gene for wrinkled pea (r) and only three genotypes are possible: RR, Rr and rr. These three genotypes produce only two phenotypes: Round (RR and Rr) and wrinkled (rr). There are no intermediate traits between round and wrinkled. If all human characteristics were controlled by simple pairs of dominant and recessive alleles like the one Mendel studied, we would have tall and short people with no intermediates. Polygenic inheritance is yet another exception to Mendel's genetic ratios.