The value of `""^(n)C_(n)+""^(n+1)C_(n)+""^(n+2)C_(n)+….+""^(n+k)C_(n)` :

Options

A. `""^(n+k+1)C_(n+1)`
B. `""^(n+k)C_(n)`
C. `""^(n+k)C_(n+1)`
D. None of these

Answer

To solve the problem of finding the value of the expression \( \binom{n}{n} + \binom{n+1}{n} + \binom{n+2}{n} + \ldots + \binom{n+k}{n} \), we will utilize the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Identify the Terms**:
The expression we need to evaluate is:
\[ \binom{n}{n} + \binom{n+1}{n} + \binom{n+2}{n} + \ldots + \binom{n+k}{n} \]

2. **Use the Property of Binomial Coefficients**:
We know the property:
\[ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \]
We will apply this property iteratively to combine the terms.

3. **Start with the First Two Terms**:
The first two terms are:
\[ \binom{n}{n} + \binom{n+1}{n} \]
Using the property, we can combine these:
\[ \binom{n}{n} + \binom{n+1}{n} = \binom{n+1}{n+1} \]

4. **Continue Combining Terms**:
Now, we will include the next term:
\[ \binom{n+2}{n} \]
So now we have:
\[ \binom{n+1}{n+1} + \binom{n+2}{n} \]
Applying the property again:
\[ \binom{n+1}{n+1} + \binom{n+2}{n} = \binom{n+2}{n+1} \]

5. **Repeat the Process**:
Continue this process for all terms up to \( \binom{n+k}{n} \):
- Combine \( \binom{n+3}{n} \) with \( \binom{n+2}{n+1} \) to get \( \binom{n+3}{n+1} \).
- Combine \( \binom{n+4}{n} \) with \( \binom{n+3}{n+1} \) to get \( \binom{n+4}{n+1} \).
- Continue this until you reach \( \binom{n+k}{n} \).

6. **Final Combination**:
After combining all terms, we will end up with:
\[ \binom{n+k+1}{n+1} \]

### Conclusion:
Thus, the final value of the expression \( \binom{n}{n} + \binom{n+1}{n} + \binom{n+2}{n} + \ldots + \binom{n+k}{n} \) is:
\[ \binom{n+k+1}{n+1} \]