`"^(30)C_(0)*^(20)C_(10)+^(31)C_(1)*^(19)C_(10)+^(32)C_(2)*18C_(10)+....^(40)C_(10)*^(10)C_(10)` is equal to

Options

A. `"^(51)C_(41)`
B. `"^(50)C_(40)`
C. `"^(51)C_(21)`
D. `"^(50)C_(40)`

Explanation

To solve the expression \( \binom{30}{0} \cdot \binom{20}{10} + \binom{31}{1} \cdot \binom{19}{10} + \binom{32}{2} \cdot \binom{18}{10} + \ldots + \binom{40}{10} \cdot \binom{10}{10} \), we can use the concept of generating functions and the Binomial Theorem.

### Step 1: Identify the pattern in the expression
The given expression consists of terms of the form \( \binom{30+k}{k} \cdot \binom{20-k}{10} \) for \( k = 0, 1, 2, \ldots, 10 \).

### Step 2: Rewrite the expression using generating functions
We can rewrite the expression as a coefficient of \( x^{10} \) in the expansion of:
\[ (1 + x)^{30} \cdot (1 + x)^{20} \]
This is because \( \binom{n}{k} \) represents the coefficient of \( x^k \) in the expansion of \( (1 + x)^n \).

### Step 3: Combine the generating functions
Thus, we have:
\[ (1 + x)^{30} \cdot (1 + x)^{20} = (1 + x)^{50} \]

### Step 4: Find the coefficient of \( x^{10} \)
We need to find the coefficient of \( x^{10} \) in \( (1 + x)^{50} \). By the Binomial Theorem, the coefficient of \( x^k \) in \( (1 + x)^n \) is given by \( \binom{n}{k} \).

So, the coefficient of \( x^{10} \) in \( (1 + x)^{50} \) is:
\[ \binom{50}{10} \]

### Step 5: Conclusion
Thus, the value of the given expression is:
\[ \binom{50}{10} \]