If `(1+x+x^(2)+x^(3))^(7)=b_(0)+b_(1)x+b^(2)x^(2)+….b_(21)x^(21)`, then find the value of
`b_(0)+b_(2)+b_(4)+……+b_(20)`

Answer

To solve the problem, we need to find the value of \( b_0 + b_2 + b_4 + \ldots + b_{20} \) from the expression \( (1 + x + x^2 + x^3)^7 \).

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ (1 + x + x^2 + x^3)^7 \]

2. **Evaluate at \( x = 1 \)**:
To find the sum of all coefficients \( b_0 + b_1 + b_2 + \ldots + b_{21} \), we substitute \( x = 1 \):
\[ (1 + 1 + 1 + 1)^7 = 4^7 \]
This gives us:
\[ b_0 + b_1 + b_2 + \ldots + b_{21} = 4^7 \]

3. **Evaluate at \( x = -1 \)**:
Next, we substitute \( x = -1 \) to find the alternating sum of the coefficients:
\[ (1 - 1 + 1 - 1)^7 = 0^7 = 0 \]
This gives us:
\[ b_0 - b_1 + b_2 - b_3 + b_4 - b_5 + \ldots + b_{21} = 0 \]

4. **Set Up the Equations**:
We now have two equations:
- From \( x = 1 \):
\[ b_0 + b_1 + b_2 + \ldots + b_{21} = 4^7 \]
- From \( x = -1 \):
\[ b_0 - b_1 + b_2 - b_3 + b_4 - b_5 + \ldots + b_{21} = 0 \]

5. **Add the Two Equations**:
Adding both equations helps eliminate the odd indexed coefficients:
\[ (b_0 + b_1 + b_2 + \ldots + b_{21}) + (b_0 - b_1 + b_2 - b_3 + \ldots + b_{21}) = 4^7 + 0 \]
This simplifies to:
\[ 2(b_0 + b_2 + b_4 + \ldots + b_{20}) = 4^7 \]

6. **Solve for \( b_0 + b_2 + b_4 + \ldots + b_{20} \)**:
Dividing both sides by 2 gives us:
\[ b_0 + b_2 + b_4 + \ldots + b_{20} = \frac{4^7}{2} = 2 \cdot 4^6 = 2^{1 + 12} = 2^{13} \]

### Final Answer:
Thus, the value of \( b_0 + b_2 + b_4 + \ldots + b_{20} \) is:
\[ \boxed{8192} \]