Which of the following is not the number of ways of selecting `n` objects from `2n` objects of which `n` objects are identical

Options

A. `2^(n)`
B. `("^(2n+1)C_(0)+^(2n+1)C_(1)+...+^(2n+1)C_(n))^(1//2)`
C. the number of possible subsets `{a_(1),a_(2),….,a_(n)}`
D. None of these

Explanation

To solve the problem of selecting `n` objects from `2n` objects where `n` objects are identical, we will analyze the different cases of selection and determine which of the given options does not represent a valid way of counting these selections.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have `2n` objects in total, out of which `n` objects are identical (let's call them type A) and the other `n` objects are distinct (let's call them type B). We need to find the number of ways to select `n` objects from these `2n` objects.

2. **Case Analysis**:
We can select `n` objects in several ways based on how many identical objects we choose:
- **Case 1**: Select `0` identical objects and `n` distinct objects. This can be done in \( \binom{n}{n} = 1 \) way.
- **Case 2**: Select `1` identical object and `n-1` distinct objects. This can be done in \( \binom{n}{n-1} = n \) ways.
- **Case 3**: Select `2` identical objects and `n-2` distinct objects. This can be done in \( \binom{n}{n-2} = \frac{n(n-1)}{2} \) ways.
- ...
- **Case n**: Select `n` identical objects and `0` distinct objects. This can be done in \( \binom{n}{0} = 1 \) way.

3. **Summing the Cases**:
The total number of ways to select `n` objects from `2n` objects can be expressed as:
\[ \text{Total Ways} = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n} \]
This is the sum of the binomial coefficients which equals \( 2^n \).

4. **Evaluating the Options**:
Now we need to check the given options to identify which one does not represent a valid way of selecting `n` objects:

- **Option A**: \( \binom{2n}{n} \) - This counts the number of ways to choose `n` objects from `2n` distinct objects, which is valid.
- **Option B**: \( \frac{1}{2} \left( \sum_{k=0}^{n} \binom{2n+1}{k} \right) \) - This expression simplifies to \( 2^{2n} \) divided by 2, which is also valid.
- **Option C**: \( 2^{2n} \) - This counts the total number of subsets of `2n` distinct objects, which is valid.

5. **Conclusion**:
Since all options A, B, and C represent valid ways of selecting `n` objects from `2n` objects, we need to identify which one is not. The question implies that one of the options does not fit the criteria.

### Final Answer:
The option that does not represent the number of ways to select `n` objects from `2n` objects where `n` are identical is **none of the options**, as all are valid.