Binomial Theorem

To determine if there is a term containing in the expansion of , we need to understand the binomial expansion of this expression.

The general term in the expansion of is given by:

Here, and , and , so our expression becomes .

The general term in the expansion is:

This implies that the power of in the general term is .

We are given that the term contains , thus we set:

Simplifying for :

This shows that is a multiple of 3. Specifically, it is a positive integral multiple since both and are non-negative integers, which means the term exists with positive integers.

Thus, the correct answer is:

is a positive integral multiple of 3.

To find the term independent of in the expansion of , we can follow these steps:

Firstly, consider the general term from the binomial expansion of :

Similarly, the general term from the binomial expansion of is:

We need to find the combined term from both expansions such that the power of is zero (i.e., independent of ). This means:

Solving the equation:

Since both and must be integers, and their sum being a non-integer indicates that there is no valid integer pair that satisfies .

Therefore, there is no term in the expansion that is independent of .

The correct answer is 0.