After simplification, what is the number of terms in the expansion of `[ ( 3x +y^(5))]^(4) - [ (3x - y )^(4) ]^(5)` ?

Options

A. 4
B. 5
C. 10
D. 11

Answer

To solve the problem, we need to find the number of terms in the expression \([(3x + y^5)^4 - (3x - y)^4]^5\) after simplification. Let's break it down step by step.

### Step 1: Simplify the expression
We start with the expression:
\[ (3x + y^5)^4 - (3x - y)^4 \]
Using the property of exponents, we can rewrite it as:
\[ (3x + y^5)^{4} - (3x - y)^{4} = (3x + y^5)^{20} - (3x - y)^{20} \]

### Step 2: Identify the number of terms in each expansion
According to the binomial theorem, the number of terms in the expansion of \((A + B)^n\) is given by \(n + 1\).

For \((3x + y^5)^{20}\):
- Here \(n = 20\), so the number of terms is:
\[ 20 + 1 = 21 \]

For \((3x - y)^{20}\):
- Similarly, \(n = 20\), so the number of terms is:
\[ 20 + 1 = 21 \]

### Step 3: Combine the expansions
Now we have:
\[ (3x + y^5)^{20} - (3x - y)^{20} \]
Both expansions have 21 terms each.

### Step 4: Analyze the cancellation of terms
When we subtract \((3x - y)^{20}\) from \((3x + y^5)^{20}\), we need to consider the nature of the terms:
- The terms in \((3x + y^5)^{20}\) will include both even and odd powers of \(y\).
- The terms in \((3x - y)^{20}\) will also include both even and odd powers of \(y\).

Since the odd-powered terms will cancel each other out (because one will be positive and the other negative), we are left with only the even-powered terms.

### Step 5: Count the remaining terms
In the expansion of \((3x + y^5)^{20}\) and \((3x - y)^{20}\), the even-powered terms will be:
- The even powers of \(y\) can be \(0, 2, 4, ..., 20\) (which gives us 11 terms).
- However, since we are only interested in the even terms that remain after cancellation, we will have half of the total terms from either expansion.

Thus, the total number of even terms remaining after cancellation is:
\[ \frac{21}{2} = 10 \text{ (since we only count even terms)} \]

### Final Answer
The total number of terms in the simplified expression is:
\[ \text{Total number of terms} = 10 \]

### Summary
The answer to the question is **10**, which corresponds to option C.