A trihybrid cross is made between two yeasts, both with genotype AaBbCc. What proportion of the offspring will be of the genotype aabbcc?

Options

A. 0
B. `(1)/(4)`
C. `(1)/(16)`
D. `(1)/(64)`

Explanation

To solve the problem of determining the proportion of offspring with the genotype aabbcc from a trihybrid cross between two yeasts with the genotype AaBbCc, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Genotype of Parents**:
The parents have the genotype AaBbCc. This means each parent can contribute either the dominant or recessive allele for each gene.

2. **Determine the Number of Gametes**:
For each gene (A, B, and C), there are two possible alleles (A or a, B or b, C or c). The number of different gametes that can be formed is given by the formula \(2^n\), where \(n\) is the number of heterozygous gene pairs.
- Here, \(n = 3\) (A, B, and C).
- Therefore, the number of gametes = \(2^3 = 8\).

3. **List the Gametes**:
The possible gametes from each parent are:
- ABC
- ABc
- AbC
- Abc
- aBC
- aBc
- abC
- abc

4. **Calculate Total Offspring**:
Since both parents can produce 8 different gametes, the total number of offspring (progeny) from the cross is:
- Total Offspring = Number of Gametes from Parent 1 × Number of Gametes from Parent 2 = \(8 \times 8 = 64\).

5. **Identify the Desired Genotype**:
We are interested in the genotype aabbcc, which is a homozygous recessive genotype for all three genes.

6. **Calculate the Probability of aabbcc**:
For each gene, the probability of getting the homozygous recessive genotype (aa, bb, cc) from a cross of AaBbCc × AaBbCc is:
- Probability of aa = 1/4 (from Aa × Aa)
- Probability of bb = 1/4 (from Bb × Bb)
- Probability of cc = 1/4 (from Cc × Cc)

Therefore, the combined probability of obtaining aabbcc is:
\[ P(aabbcc) = P(aa) \times P(bb) \times P(cc) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64} \]

7. **Conclusion**:
Since there are 64 total offspring, the expected number of offspring with the genotype aabbcc is:
- Number of aabbcc = Total Offspring × Probability of aabbcc = \(64 \times \frac{1}{64} = 1\).

Thus, the proportion of offspring with the genotype aabbcc is:
\[ \frac{1}{64} \]

### Final Answer:
The proportion of offspring with the genotype aabbcc is \( \frac{1}{64} \).

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