The value of `((100),(0))((200),(150))+((100),(1))((200),(151))+......+((100),(50))((200),(200))` equals (where `((n),(r ))="^(n)C_(r)`)

Options

A. `((300),(50))`
B. `((100),(50))((200),(150))`
C. `((100),(50))^(2)`
D. `((300),(50))^(2)`

Explanation

To solve the problem, we need to evaluate the expression:

\[ \sum_{r=0}^{50} \binom{100}{r} \binom{200}{150+r} \]

This expression represents the sum of products of binomial coefficients. We can interpret this using the binomial theorem.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression can be rewritten as:
\[ \sum_{r=0}^{50} \binom{100}{r} \binom{200}{150+r} \]
This means we are summing the products of binomial coefficients for \( r \) ranging from 0 to 50.

2. **Using the Binomial Theorem**:
According to the binomial theorem, we know that:
\[ (1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
We can apply this theorem to find the coefficients we need.

3. **Setting Up the Binomial Expansion**:
We can express the required sum using the binomial expansion:
\[ (1+x)^{100} \cdot (1+x)^{200} = (1+x)^{300} \]
The coefficient of \( x^{50} \) in \( (1+x)^{300} \) gives us the sum we are looking for.

4. **Finding the Coefficient**:
The coefficient of \( x^{50} \) in \( (1+x)^{300} \) is given by:
\[ \binom{300}{50} \]

5. **Final Calculation**:
Therefore, the value of the original expression is:
\[ \binom{300}{50} \]

6. **Conclusion**:
The final answer is:
\[ \binom{300}{50} = 300C50 \]