A polygenic trait is controlled by three genes A, B, and C. In a cross AaBbCc x AaBbCc, the offspring show seven phenotypic classes in the ratio 1 : 6 : m : 20 : m : 6 : 1. What is the value of m for the intermediate phenotypes?

Introduction / Context:
Polygenic inheritance involves traits that are controlled by several genes, each contributing a small additive effect to the phenotype. Examples include human height, skin color, and kernel color in some crops. When multiple gene pairs influence a single trait, crosses between heterozygotes can produce a wide range of phenotypes that follow predictable ratios. This question asks you to determine a missing term in a seven class phenotypic ratio for a trait controlled by three gene pairs.

Given Data / Assumptions:

• The trait is controlled by three independently assorting genes A, B, and C.
• Each gene has two alleles, with capital letters usually contributing to a dominant or additive effect.
• The cross is AaBbCc x AaBbCc, meaning both parents are heterozygous at all three loci.
• The observed phenotypic ratio is 1 : 6 : m : 20 : m : 6 : 1, representing seven phenotypic classes.
• We assume equal and additive contribution from each dominant allele and no gene interaction beyond additive effects.

Concept / Approach:
For a polygenic trait controlled by n gene pairs with additive effects, the number of possible phenotypic classes is 2n + 1. With three genes, there are 2*3 + 1 = 7 phenotypic classes, which matches the seven terms given. The distribution of phenotypes can be derived from counting how many dominant alleles are present in the genotype. In this case, dominant alleles from A, B, and C can sum from zero to six. The coefficients in the ratio correspond to the number of ways of obtaining each total number of dominant alleles in the offspring of Aa x Aa for each gene. This distribution is the expansion of (1 : 2 : 1)^3, which yields a known pattern.

Step-by-Step Solution:
Step 1: For a single gene pair Aa x Aa, the genotypic ratio is 1 AA : 2 Aa : 1 aa. In terms of number of dominant alleles, this corresponds to 2 dominant alleles (AA), 1 dominant allele (Aa), and 0 dominant alleles (aa). Step 2: Because A, B, and C assort independently, the combined distribution of dominant alleles from three gene pairs is given by the expansion of (1 : 2 : 1)^3. Step 3: Expand (1 + 2 + 1)^3 first to check the total number of outcomes: (4)^3 = 64, which matches the total number of genotypes possible in a AaBbCc x AaBbCc cross. Step 4: The detailed expansion of (1 : 2 : 1)^3 yields the sequence 1 : 6 : 15 : 20 : 15 : 6 : 1, corresponding to offspring with 0, 1, 2, 3, 4, 5, and 6 dominant alleles respectively. Step 5: Compare this to the given phenotypic ratio 1 : 6 : m : 20 : m : 6 : 1. The terms at positions 3 and 5 must both be 15 to match the known expansion. Step 6: Therefore the missing value m is 15.

Verification / Alternative check:
Another way to verify the result is to recognize that the distribution of the number of dominant alleles follows a multinomial distribution. You could count combinations leading to a given number of dominant alleles by considering permutations of AA, Aa, and aa genotypes across three genes and using combinations and coefficients. When you systematically count the possible genotypes with two dominant alleles or four dominant alleles, you would find 15 each, in agreement with the (1 : 2 : 1)^3 expansion. Finally, if you sum the full sequence 1 + 6 + 15 + 20 + 15 + 6 + 1, you obtain 64, which matches the expected total number of offspring genotypes.

Why Other Options Are Wrong:
Option A: Ten does not fit the standard expansion for three gene pairs and would produce a total sum less than 64 when inserted into the ratio.

Option C: Eighteen is inconsistent with the binomial based distribution from three heterozygous gene pairs and would distort the expected symmetrical pattern around the central class.

Option D: Twenty four is too large and would break the symmetrical nature of the phenotypic distribution, which must mirror around the middle class.

Common Pitfalls:
A common error is to treat polygenic inheritance like simple Mendelian inheritance and ignore the distribution of additive effects across multiple loci. Another pitfall is to misremember the standard 1 : 4 : 6 : 4 : 1 pattern from binomial expansions and apply it incorrectly. For three gene pairs, it is useful to memorize or re derive the full 1 : 6 : 15 : 20 : 15 : 6 : 1 ratio. Remember that in polygenic traits with additive effects, counting the total number of dominant alleles is the key step.

Final Answer:
The missing value in the phenotypic ratio is 15, so the full ratio is 1 : 6 : 15 : 20 : 15 : 6 : 1.