If `(1 + x + x^(2) + x^(3))^(n) = a_(0) + a_(1)x + a_(2)x^(2)+"……….."a_(3n)x^(3n)` then which of following are correct

Options

A. `a_(0) + a_(1) + a_(2) +……+a_(3n) = 2^(2n)`
B. `a_(0) + a_(2) + a_(4) +"……"= a_(1) + a_(3) + a_(5) +"…….."`
C. `a_(0) = a_(3n), a_(1) = a_(3n - 1), a_(2) = a_(3n - 2)`
D. `a_(0) + a_(2) + a_(4) + "……." = 2^(2n - 1)`

Explanation

To solve the problem, we start with the expression given:

\[ (1 + x + x^2 + x^3)^n = a_0 + a_1 x + a_2 x^2 + \ldots + a_{3n} x^{3n} \]

We need to analyze the coefficients \(a_k\) for different values of \(x\) to determine which options are correct.

### Step 1: Substitute \(x = 1\)

Substituting \(x = 1\) into the equation:

\[ (1 + 1 + 1^2 + 1^3)^n = (1 + 1 + 1 + 1)^n = 4^n \]

The right-hand side becomes:

\[ a_0 + a_1 + a_2 + \ldots + a_{3n} \]

Thus, we have:

\[ a_0 + a_1 + a_2 + \ldots + a_{3n} = 4^n \]

### Step 2: Substitute \(x = -1\)

Now, substituting \(x = -1\):

\[ (1 - 1 + (-1)^2 + (-1)^3)^n = (1 - 1 + 1 - 1)^n = 0^n = 0 \]

The right-hand side becomes:

\[ a_0 - a_1 + a_2 - a_3 + a_4 - \ldots + (-1)^{3n} a_{3n} \]

This means:

\[ a_0 + a_2 + a_4 + \ldots = a_1 + a_3 + a_5 + \ldots \]

### Step 3: Analyze the results

From Step 1, we have:

\[ a_0 + a_1 + a_2 + \ldots + a_{3n} = 4^n \]

From Step 2, we have:

\[ a_0 + a_2 + a_4 + \ldots = a_1 + a_3 + a_5 + \ldots \]

Let \(S_e\) be the sum of the coefficients of even powers and \(S_o\) be the sum of the coefficients of odd powers:

1. \(S_e + S_o = 4^n\)
2. \(S_e = S_o\)

From these equations, we can conclude:

\[ 2S_e = 4^n \implies S_e = 2^{2n} \quad \text{and} \quad S_o = 2^{2n} \]

### Step 4: Verify the options

1. **Option 1**: \(a_0 + a_1 + a_2 + \ldots + a_{3n} = 4^n\) - **Correct**
2. **Option 2**: \(a_0 + a_2 + a_4 + \ldots = a_1 + a_3 + a_5 + \ldots\) - **Correct**
3. **Option 3**: \(a_0 = a_3\) and \(a_1 = a_2\) - **Incorrect** (no relation derived)
4. **Option 4**: \(a_0 + a_2 + a_4 + \ldots = 2^{2n - 1}\) - **Correct** (since \(S_e = 2^{2n}\))

### Final Conclusion

The correct options are 1, 2, and 4.