The value of `sum_(r=0)^50 (.^(100)C_r.^(200)C_(150+r))` is equal to

Options

A. `""^(300)C_(50)`
B. `""^(100)C_(50)*""^(200)C_(150)`
C. `""^(100)C_(50)`
D. None of these

Answer

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

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

### Step 1: Understand the Binomial Coefficients
The binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) successes in \(n\) trials. In our case, we have two binomial coefficients: one from \(100\) and the other from \(200\).

### Step 2: Apply the Binomial Theorem
We can use the Binomial Theorem, which states that:

\[ (1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]

We will apply this theorem to both \( (1+x)^{100} \) and \( (1+x)^{200} \).

### Step 3: Expand the Binomials
1. Expand \( (1+x)^{100} \):
\[ (1+x)^{100} = \sum_{r=0}^{100} \binom{100}{r} x^r \]

2. Expand \( (1+x)^{200} \):
\[ (1+x)^{200} = \sum_{s=0}^{200} \binom{200}{s} x^s \]

### Step 4: Combine the Expansions
Now, we want to find the coefficient of \(x^{50}\) in the product of these two expansions:

\[ (1+x)^{100} \cdot (1+x)^{200} = (1+x)^{300} \]

### Step 5: Identify the Coefficient
The coefficient of \(x^{50}\) in \((1+x)^{300}\) can be found using the binomial coefficient:

\[ \text{Coefficient of } x^{50} \text{ in } (1+x)^{300} = \binom{300}{50} \]

### Conclusion
Thus, the value of the sum \(S\) is:

\[ S = \binom{300}{50} \]

### Final Answer
The value of \( \sum_{r=0}^{50} \binom{100}{r} \binom{200}{150+r} \) is equal to \( \binom{300}{50} \).

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