Appearance of which of the following phenotypic ratio indicates polygenic inheritance?
(a) `1:2:1`
(b) `12:3:1`
(c) `1 : 4 : 6 : 4 : 1 `
(d) `27 : 9 : 9 : 9 : 3 : 3 : 3: 1`

Options

A. `1:2:1`
B. `12:3:1`
C. `1 : 4 : 6 : 4 : 1 `
D. `27 : 9 : 9 : 9 : 3 : 3 : 3: 1`

Answer

To determine which phenotypic ratio indicates polygenic inheritance, we will analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding Polygenic Inheritance**:
- Polygenic inheritance refers to a trait that is controlled by multiple genes. This results in a continuous range of phenotypes rather than discrete categories.
- The phenotypic ratios resulting from polygenic inheritance are typically more complex and can show a bell-shaped curve when plotted.

2. **Analyzing the Options**:
- **(a) 1:2:1**:
- This ratio typically indicates a monohybrid cross involving a single gene with two alleles. This is not indicative of polygenic inheritance.

- **(b) 12:3:1**:
- This ratio is characteristic of a dihybrid cross involving epistasis, where one gene affects the expression of another. This does not represent polygenic inheritance.

- **(c) 1:4:6:4:1**:
- This ratio is indicative of polygenic inheritance. It suggests that multiple genes are contributing to the trait, with a range of phenotypes appearing in a specific pattern. The numbers suggest that there are multiple genotypes producing the same phenotype.

- **(d) 27:9:9:9:3:3:3:1**:
- This ratio is typical of a trihybrid cross and involves multiple genes but does not represent the continuous variation seen in polygenic inheritance.

3. **Conclusion**:
- Among the options, **(c) 1:4:6:4:1** is the phenotypic ratio that indicates polygenic inheritance. This ratio reflects the cumulative effect of multiple genes on a single trait.

### Final Answer:
The correct answer is **(c) 1:4:6:4:1**.