If the sum of all the coefficients in the expansion of `(1-3x+10x^(2))^(n)` is `alpha` and the sum of all the coefficients in the expansion of `(1+x^(2))^(n)` is `beta`, then which of the following is correct ?

Options

A. `alpha= 3 beta`
B. `alpha= beta^(3)`
C. `beta= alpha^(3)`
D. `alpha = 2beta`

Answer

To solve the problem, we need to find the sums of the coefficients in the expansions of the given expressions and establish a relationship between them.

### Step-by-Step Solution:

1. **Identify the Expressions**:
We have two expressions:
- \( (1 - 3x + 10x^2)^n \)
- \( (1 + x^2)^n \)

2. **Find the Sum of Coefficients for the First Expression**:
To find the sum of the coefficients in the expansion of \( (1 - 3x + 10x^2)^n \), we substitute \( x = 1 \):
\[ \text{Sum of coefficients} = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = (8)^n \]
Thus, we have:
\[ \alpha = 8^n \]

3. **Find the Sum of Coefficients for the Second Expression**:
For the second expression \( (1 + x^2)^n \), we also substitute \( x = 1 \):
\[ \text{Sum of coefficients} = (1 + 1^2)^n = (1 + 1)^n = (2)^n \]
Thus, we have:
\[ \beta = 2^n \]

4. **Establish the Relationship Between Alpha and Beta**:
We have:
\[ \alpha = 8^n \quad \text{and} \quad \beta = 2^n \]
We can express \( 8^n \) in terms of \( 2^n \):
\[ 8^n = (2^3)^n = (2^n)^3 \]
Therefore, we can write:
\[ \alpha = \beta^3 \]

### Conclusion:
The relationship between \( \alpha \) and \( \beta \) is:
\[ \alpha = \beta^3 \]