Binomial Theorem

Step 1: Understand the binomial expansion.
The binomial expansion of is given by , where are the binomial coefficients. The sum can be interpreted as a specific sum of terms from this expansion.

Step 2: Calculate the sum.
Using the properties of the binomial expansion, the value of the given sum is . This result comes from evaluating the series formed by the coefficients and their corresponding factors.

Step 3: Conclusion.
Thus, the correct answer is , which corresponds to option (B).

Step 1: Identify the multinomial expansion.
In the expansion of , the number of terms corresponds to the number of distinct combinations of powers of in the expansion. Each term in the expansion is of the form , where and .
Step 2: Apply the stars and bars method.
The number of distinct non-negative integer solutions to the equation is given by the formula for combinations with repetition:
Step 3: Conclusion.
Thus, the number of terms in the expansion is 21, which corresponds to option (C).