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

Options

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

Explanation

To find the value of \( r \) for which the expression

\[ \binom{20}{r} \binom{20}{0} + \binom{20}{r-1} \binom{20}{1} + \binom{20}{r-2} \binom{20}{2} + \ldots + \binom{20}{0} \binom{20}{r} \]

is maximized, we can follow these steps:

### Step 1: Understand the Expression
The expression can be interpreted as the coefficients of \( x^r \) in the expansion of \( (1 + x)^{20} \cdot (1 + x)^{20} \).

### Step 2: Rewrite the Expression
Using the binomial theorem, we can rewrite the expression as:

\[ (1 + x)^{20} \cdot (1 + x)^{20} = (1 + x)^{40} \]

### Step 3: Identify the Coefficient
The coefficient of \( x^r \) in \( (1 + x)^{40} \) is given by \( \binom{40}{r} \).

### Step 4: Find the Maximum Coefficient
To maximize \( \binom{40}{r} \), we use the property that the binomial coefficients are maximized when \( r \) is approximately \( \frac{n}{2} \). Here, \( n = 40 \).

### Step 5: Calculate \( r \)
Thus, the value of \( r \) that maximizes \( \binom{40}{r} \) is:

\[ r = \frac{40}{2} = 20 \]

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

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