If `m, n, r, in N` then `.^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0)`
`=` coefficient of `x^(r)` in `(1+x)^(m)(1+x)^(n)`
`=` coefficient of `x^(f)` in `(1+x)^(m+n)`
The value of r for which `S = .^(20)C_(r.).^(10)C_(0)+.^(20)C_(r-1).^(10)C_(1)+"........".^(20)C_(0).^(10)C_(r)` is maximum can not be

Options

A. `7`
B. `8`
C. `10`
D. `15`

Explanation

To solve the problem, we need to analyze the expression given and find the value of \( r \) for which the sum \( S \) is maximum. Let's break it down step by step.

### Step 1: Understand the Expression
The expression we need to analyze is:
\[ S = \binom{20}{r} \cdot \binom{10}{0} + \binom{20}{r-1} \cdot \binom{10}{1} + \ldots + \binom{20}{0} \cdot \binom{10}{r} \]
This expression represents the sum of products of binomial coefficients.

### Step 2: Relate to Binomial Expansion
From the properties of binomial coefficients, we know that:
\[ S = \text{coefficient of } x^r \text{ in } (1+x)^{20} \cdot (1+x)^{10} = \text{coefficient of } x^r \text{ in } (1+x)^{30} \]
This means we can express \( S \) as:
\[ S = \binom{30}{r} \]
where \( \binom{30}{r} \) is the binomial coefficient representing the coefficient of \( x^r \) in the expansion of \( (1+x)^{30} \).

### Step 3: Find Maximum Value
The binomial coefficient \( \binom{30}{r} \) reaches its maximum value when \( r \) is around \( \frac{30}{2} = 15 \). This is due to the symmetry and properties of binomial coefficients.

### Step 4: Determine Values of \( r \)
Since \( r \) must be a natural number, we can have \( r = 15 \) as the maximum point. The values of \( r \) that will yield maximum \( S \) are \( 14, 15, \) and \( 16 \).

### Step 5: Identify the Value of \( r \) That Cannot Be Maximum
The question asks for the value of \( r \) for which \( S \) cannot be maximum. Since \( r \) can take values around 15 for maximum \( S \), any value outside the range of \( 14, 15, 16 \) cannot yield a maximum.

Thus, the value of \( r \) that cannot be maximum is any integer not equal to \( 14, 15, \) or \( 16 \). For instance, \( r = 0 \) or \( r = 30 \) would not give maximum \( S \).

### Final Answer
The value of \( r \) for which \( S \) is maximum cannot be:
\[ \text{Any value not in } \{14, 15, 16\}, \text{ for example, } r = 0 \text{ or } r = 30. \]