A trihybrid cross is made between two plants with genotypes A/a B/b C/c how many offspring of such cross will have a genotype a/a b/b c/c-

Options

A. `1//64`
B. `1//4`
C. `1//16`
D. `1//32`

Answer

To solve the problem of how many offspring will have the genotype a/a b/b c/c from a trihybrid cross between two plants with the genotypes A/a B/b C/c, we can follow these steps:

### Step 1: Identify the Genotypes
The parent plants have the genotype A/a B/b C/c. This means each plant is heterozygous for three traits (A, B, and C).

### Step 2: Determine the Number of Gametes
For each gene, the heterozygous genotype (A/a, B/b, C/c) can produce two types of gametes:
- For A/a: Gametes are A and a
- For B/b: Gametes are B and b
- For C/c: Gametes are C and c

Since there are three genes, the total number of different gametes that can be produced is calculated as:
\[ \text{Number of gametes} = 2^n \]
where \( n \) is the number of heterozygous traits. Here, \( n = 3 \).

Thus, the number of gametes is:
\[ 2^3 = 8 \]

### Step 3: Calculate the Total Offspring
When two plants are crossed, the total number of offspring produced can be calculated as:
\[ \text{Total offspring} = \text{Number of gametes from plant 1} \times \text{Number of gametes from plant 2} \]
Since both plants can produce 8 types of gametes:
\[ \text{Total offspring} = 8 \times 8 = 64 \]

### Step 4: Determine the Probability of the Recessive Genotype
To find the probability of obtaining the genotype a/a b/b c/c, we need to consider the combinations that lead to this genotype:
- For a/a: The probability of getting a from each parent is \( \frac{1}{2} \) (since each parent can contribute either A or a).
- For b/b: Similarly, the probability of getting b from each parent is \( \frac{1}{2} \).
- For c/c: The probability of getting c from each parent is \( \frac{1}{2} \).

Thus, the combined probability of obtaining the genotype a/a b/b c/c is:
\[ P(a/a) \times P(b/b) \times P(c/c) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

### Step 5: Calculate the Expected Number of Offspring with the Recessive Genotype
Now, to find the expected number of offspring with the genotype a/a b/b c/c, we multiply the total number of offspring by the probability of the recessive genotype:
\[ \text{Expected number of a/a b/b c/c} = \text{Total offspring} \times P(a/a b/b c/c) \]
\[ = 64 \times \frac{1}{8} = 8 \]

### Step 6: Find the Fraction of Offspring with the Recessive Genotype
Finally, to express this as a fraction of the total offspring:
\[ \text{Fraction} = \frac{8}{64} = \frac{1}{8} \]

### Conclusion
The probability of obtaining an offspring with the genotype a/a b/b c/c in this trihybrid cross is \( \frac{1}{64} \).