`(1)/(4) T T, (1)/(2)Tt,(1)/(4) t t` is binomial expansion of

Options

A. `((1)/(2)T+(1)/(2)t)^(2)`
B. `((1)/(4)T+(1)/(4)t)^(2)`
C. `((1)/(4)T+(1)/(2)t)^(2)`
D. `((1)/(2)T+(1)/(4)t)^(2)`

Answer

To solve the problem where we need to identify the binomial expansion represented by the ratios \( \frac{1}{4} TT, \frac{1}{2} Tt, \frac{1}{4} tt \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Ratios**:
The given ratios \( \frac{1}{4} TT, \frac{1}{2} Tt, \frac{1}{4} tt \) represent the probabilities of different genotypes resulting from a genetic cross.

2. **Identifying the Genotypes**:
- \( TT \) represents the homozygous dominant genotype.
- \( Tt \) represents the heterozygous genotype.
- \( tt \) represents the homozygous recessive genotype.

3. **Setting Up the Binomial Expression**:
We recognize that these ratios can be derived from a binomial expansion of the form \( (p + q)^2 \), where \( p \) is the probability of one allele and \( q \) is the probability of the other allele.

4. **Finding the Probabilities**:
To find \( p \) and \( q \):
- The total probability must equal 1.
- Given the ratios, we can set \( p = \frac{1}{2}T \) and \( q = \frac{1}{2}t \).

5. **Using the Binomial Expansion Formula**:
The binomial expansion of \( (p + q)^2 \) is given by:
\[ (p + q)^2 = p^2 + 2pq + q^2 \]
Substituting \( p = \frac{1}{2}T \) and \( q = \frac{1}{2}t \):
\[ \left(\frac{1}{2}T + \frac{1}{2}t\right)^2 = \left(\frac{1}{2}T\right)^2 + 2\left(\frac{1}{2}T\right)\left(\frac{1}{2}t\right) + \left(\frac{1}{2}t\right)^2 \]
This simplifies to:
\[ \frac{1}{4}TT + \frac{1}{2}Tt + \frac{1}{4}tt \]

6. **Conclusion**:
Therefore, the binomial expansion that corresponds to the given ratios is:
\[ \left(\frac{1}{2}T + \frac{1}{2}t\right)^2 \]

### Final Answer:
The binomial expansion represented by \( \frac{1}{4} TT, \frac{1}{2} Tt, \frac{1}{4} tt \) is \( \left(\frac{1}{2}T + \frac{1}{2}t\right)^2 \).