Binomial Theorem

The given expression is:

Each term in the expansion arises from the multinomial expansion of . Let:

The multinomial expansion of is:

Substitute back :

Step 1: Match powers of .

We need:

Solve these equations simultaneously:

From , substitute into :

Substitute into :

Using , find and :

Step 2: Coefficient of the term.

The coefficient is given by the multinomial coefficient:

Simplify:

Conclusion:

The coefficient of is:

1. Step 1: The given series can be expressed as:

2. Step 2: Split the summation into two parts:

3. Step 3: Use the binomial summation properties:

4. Step 4: Substitute these results:

5. Step 5: Simplify:

6. Additionally: Consider , then

Substituting , we also get the same result.

Thus, the correct answers are (A) and (C).

The given expression is:

We are tasked to compute:

Step 1: Simplify . Let:

This is a finite geometric series:

Thus, the given expression becomes:

Step 2: Expand the numerator and denominator. Expand the expression:

Use the binomial theorem to expand and :

Step 3: Coefficients of . The coefficient corresponds to the term in the product:

To extract the coefficient of , combine terms where . For this:

The coefficient of is:

Step 4: Compute . Substitute into:

Rearrange the summations:

The inner summation evaluates to 0 due to alternating signs, as ensures cancellation of terms.

Conclusion: The value of is: