The value of r for which
`.^(20)C_(r ), .^(20)C_(r - 1) .^(20)C_(1) + .^(20)C_(2) + …… + .^(20)C_(0) .^(20)C_(r )` is maximum, is

Options

A. 20
B. 15
C. 11
D. 10

Answer

To solve the problem, we need to find the value of \( r \) for which the expression

\[ \binom{20}{r} \cdot \binom{20}{r-1} \cdot \sum_{k=0}^{2} \binom{20}{k} \]

is maximized.

### Step-by-step Solution:

1. **Understanding the Expression**:
The expression can be rewritten using the binomial theorem. The sum \( \sum_{k=0}^{2} \binom{20}{k} \) represents the sum of the first three binomial coefficients from the expansion of \( (1+x)^{20} \).

2. **Using the Binomial Theorem**:
According to the binomial theorem, we have:
\[ (1+x)^{20} = \sum_{k=0}^{20} \binom{20}{k} x^k \]
Therefore, we can express our original sum in terms of the coefficients of \( (1+x)^{20} \).

3. **Combining the Terms**:
The expression can be interpreted as the coefficient of \( x^r \) in the expansion of:
\[ (1+x)^{20} \cdot (1+x)^{20} = (1+x)^{40} \]
This means we are looking for the coefficient of \( x^r \) in \( (1+x)^{40} \).

4. **Finding the Coefficient**:
The coefficient of \( x^r \) in \( (1+x)^{40} \) is given by \( \binom{40}{r} \).

5. **Maximizing the Coefficient**:
The binomial coefficient \( \binom{40}{r} \) is maximized when \( r \) is around \( \frac{40}{2} = 20 \). This is a property of binomial coefficients, where they reach their maximum value at \( n/2 \).

6. **Conclusion**:
Therefore, the value of \( r \) for which the expression is maximized is:
\[ r = 20 \]

### Final Answer:
The value of \( r \) for which the expression is maximum is \( \boxed{20} \).