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(0 le r le 30)` for which S = `.^(20)C_(r).^(10)C_(0) + .^(20)C_(r-1).^(10)C_(1) + ........ + .^(20)C_(0).^(10)C_(r)` is minimum can not be

Options

A. `0`
B. `1`
C. `30`
D. `15`

Explanation

To solve the problem, we need to analyze the expression given and find the values of \( r \) for which the sum \( S \) is minimized. The expression for \( S \) is:

\[ S = \binom{20}{r} \binom{10}{0} + \binom{20}{r-1} \binom{10}{1} + \ldots + \binom{20}{0} \binom{10}{r} \]

This sum can be interpreted as the coefficient of \( x^r \) in the expansion of \( (1+x)^{20} (1+x)^{10} \), which simplifies to \( (1+x)^{30} \).

### Step-by-Step Solution:

1. **Understanding the Coefficient**:
The coefficient of \( x^r \) in \( (1+x)^{30} \) is given by \( \binom{30}{r} \).

2. **Finding Minimum Values**:
We need to find the values of \( r \) where \( S \) is minimized. The binomial coefficient \( \binom{30}{r} \) reaches its minimum values at the extremes of \( r \), specifically at \( r = 0 \) and \( r = 30 \).

3. **Identifying Non-Minimum Values**:
Since \( S \) is minimized at \( r = 0 \) and \( r = 30 \), the values of \( r \) for which \( S \) is not minimum are all other integers from \( 1 \) to \( 29 \).

4. **Conclusion**:
Therefore, the values of \( r \) for which \( S \) cannot be minimum are:

\[ r \in \{1, 2, 3, \ldots, 29\} \]

### Final Answer:
The values of \( r \) for which \( S \) cannot be minimum are all integers from \( 1 \) to \( 29 \).

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