In the expansion of `(x + a)^(n)` the sum of even
terms is E and that of odd terms is O, then OE is equal to

Options

A. `(x +a)^(2n) - (x - a)^(2n)`
B. `(1)/(4) {(x +a)^(2n) - (x - a)^(2n)}`
C. `(1)/(4) {(x +a)^(2n) + (x - a)^(2n)}`
D. none of these

Explanation

To solve the problem, we need to find the product of the sums of the even and odd terms (E and O) in the expansion of \((x + a)^n\).

### Step-by-Step Solution:

1. **Understanding the Expansion**:
The binomial expansion of \((x + a)^n\) can be expressed as:
\[ (x + a)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \ldots + C_n x^n \]
where \(C_k\) are the binomial coefficients.

2. **Identifying Even and Odd Terms**:
- The even terms are \(C_0, C_2, C_4, \ldots\) (terms with even powers of \(x\)).
- The odd terms are \(C_1, C_3, C_5, \ldots\) (terms with odd powers of \(x\)).
- Let \(E\) be the sum of the even terms and \(O\) be the sum of the odd terms.

3. **Using the Binomial Theorem**:
We can express the sums of even and odd terms using the expansions of \((x + a)^n\) and \((x - a)^n\):
\[ (x + a)^n = E + O \]
\[ (x - a)^n = E - O \]

4. **Setting Up the Equations**:
From the above, we can set up two equations:
- Equation 1: \(E + O = (x + a)^n\)
- Equation 2: \(E - O = (x - a)^n\)

5. **Solving for E and O**:
To find \(E\) and \(O\), we can add and subtract these equations:
- Adding the two equations:
\[ 2E = (x + a)^n + (x - a)^n \]
Thus,
\[ E = \frac{(x + a)^n + (x - a)^n}{2} \]
- Subtracting the second equation from the first:
\[ 2O = (x + a)^n - (x - a)^n \]
Thus,
\[ O = \frac{(x + a)^n - (x - a)^n}{2} \]

6. **Finding the Product OE**:
Now we can find the product \(OE\):
\[ OE = E \cdot O = \left(\frac{(x + a)^n + (x - a)^n}{2}\right) \cdot \left(\frac{(x + a)^n - (x - a)^n}{2}\right) \]
This simplifies to:
\[ OE = \frac{1}{4} \left((x + a)^n + (x - a)^n\right) \left((x + a)^n - (x - a)^n\right) \]

7. **Using the Difference of Squares**:
The expression can be simplified further using the difference of squares:
\[ OE = \frac{1}{4} \left((x + a)^{2n} - (x - a)^{2n}\right) \]

### Final Result:
Thus, we conclude that:
\[ OE = \frac{1}{4} \left((x + a)^{2n} - (x - a)^{2n}\right) \]