If `(1+x+x^2+x^3)^100=a_0+a_1x+a_2x^2+.......+a_300x^300,` then

Options

A. `a_(1)=100`
B. `a_(0)+a_(1)+a_(2)+…..+a_(300)` is divisible by 1024
C. coefficients equidistant from beginning and end are equal
D. `a_(0)+a_(2)+a_(4)+……..+a_(300)=a_(1)+a_(3)+a_(5)+……+a_(299)`

Answer

To solve the problem, we need to analyze the expression \( (1 + x + x^2 + x^3)^{100} \) and find the coefficients \( a_0, a_1, a_2, \ldots, a_{300} \) in the expansion.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression can be rewritten using the formula for the sum of a geometric series:
\[ 1 + x + x^2 + x^3 = \frac{1 - x^4}{1 - x} \]
Therefore, we can express our original expression as:
\[ (1 + x + x^2 + x^3)^{100} = \left(\frac{1 - x^4}{1 - x}\right)^{100} = (1 - x^4)^{100} \cdot (1 - x)^{-100} \]

2. **Using Binomial Theorem**:
We can apply the Binomial Theorem to expand both parts:
- For \( (1 - x^4)^{100} \):
\[ (1 - x^4)^{100} = \sum_{k=0}^{100} \binom{100}{k} (-1)^k x^{4k} \]
- For \( (1 - x)^{-100} \):
\[ (1 - x)^{-100} = \sum_{m=0}^{\infty} \binom{m + 99}{99} x^m \]

3. **Finding Coefficients**:
The coefficient \( a_n \) of \( x^n \) in the expansion of \( (1 + x + x^2 + x^3)^{100} \) can be found by combining the two expansions:
\[ a_n = \sum_{k=0}^{\lfloor n/4 \rfloor} \binom{100}{k} (-1)^k \binom{n - 4k + 99}{99} \]
This formula gives us the coefficients \( a_n \) for \( n = 0, 1, 2, \ldots, 300 \).

4. **Finding Specific Coefficients**:
- To find \( a_1 \):
\[ a_1 = \sum_{k=0}^{\lfloor 1/4 \rfloor} \binom{100}{k} (-1)^k \binom{1 - 4k + 99}{99} \]
Since \( k = 0 \) is the only valid term:
\[ a_1 = \binom{100}{0} (-1)^0 \binom{100}{99} = 1 \cdot 1 \cdot 100 = 100 \]

5. **Summation of Coefficients**:
The total sum of the coefficients \( a_0 + a_1 + a_2 + \ldots + a_{300} \) can be found by evaluating the expression at \( x = 1 \):
\[ (1 + 1 + 1 + 1)^{100} = 4^{100} \]

6. **Checking for Equidistance**:
Since the polynomial is symmetric, the coefficients equidistant from the beginning and end are equal. This means \( a_k = a_{300-k} \).

7. **Divisibility Check**:
To check if \( 2199 \) is divisible by \( 1024 \):
- \( 2199 \) is not divisible by \( 1024 \) since \( 1024 \) is \( 2^{10} \) and \( 2199 \) is not a multiple of \( 1024 \).

### Conclusion:
From the analysis, we find:
- \( a_1 = 100 \) (Option A is correct)
- The sum of coefficients \( a_0 + a_1 + \ldots + a_{300} = 4^{100} \) (Option B is correct)
- Coefficients equidistant from beginning and end are equal (Option C is correct)
- \( 2199 \) is not divisible by \( 1024 \) (Option D is incorrect)

Thus, the correct options are A, B, and C.