Let `(1+x+x^(2))^(5)=a_(0)+a_(1)x+a_(2)x^(2)+…….+a_(10)x^(10)` then value of `a_(1)+a_(4)+a_(7)+a_(10)` is ……

Explanation

To solve the problem, we need to find the coefficients \( a_1 + a_4 + a_7 + a_{10} \) from the expansion of \( (1 + x + x^2)^5 \).

### Step-by-step Solution:

1. **Understanding the Expression**:
We start with the expression \( (1 + x + x^2)^5 \). This can be expanded using the multinomial theorem, but we will use a different approach to find specific coefficients.

2. **Finding the Coefficients**:
We can evaluate the expression at specific values of \( x \) to find the coefficients we need. The roots of the equation \( 1 + x + x^2 = 0 \) are the cube roots of unity, denoted as \( \omega \) and \( \omega^2 \), where \( \omega = e^{2\pi i / 3} \) and \( \omega^3 = 1 \).

3. **Substituting Values**:
We will substitute \( x = 1 \), \( x = \omega \), and \( x = \omega^2 \) into the expression \( (1 + x + x^2)^5 \).

- **Substituting \( x = 1 \)**:
\[ (1 + 1 + 1^2)^5 = 3^5 = 243 \]
This gives us the sum of all coefficients:
\[ a_0 + a_1 + a_2 + \ldots + a_{10} = 243 \]

- **Substituting \( x = \omega \)**:
\[ (1 + \omega + \omega^2)^5 = 0^5 = 0 \]
This gives us:
\[ a_0 + a_1 \omega + a_2 \omega^2 + a_3 \omega^3 + a_4 \omega^4 + a_5 \omega^5 + a_6 \omega^6 + a_7 \omega^7 + a_8 \omega^8 + a_9 \omega^9 + a_{10} \omega^{10} = 0 \]
Since \( \omega^3 = 1 \), we can simplify this to:
\[ a_0 + a_1 \omega + a_2 \omega^2 + a_3 + a_4 \omega + a_5 \omega^2 + a_6 + a_7 \omega + a_8 \omega^2 + a_9 + a_{10} \omega = 0 \]

- **Substituting \( x = \omega^2 \)**:
\[ (1 + \omega^2 + \omega)^5 = 0^5 = 0 \]
This gives us:
\[ a_0 + a_1 \omega^2 + a_2 \omega + a_3 + a_4 \omega^2 + a_5 \omega + a_6 + a_7 \omega^2 + a_8 \omega + a_9 + a_{10} \omega^2 = 0 \]

4. **Setting Up the System of Equations**:
We now have three equations:
- \( a_0 + a_1 + a_2 + \ldots + a_{10} = 243 \) (Equation 1)
- \( a_0 + a_3 + a_6 + a_9 + (a_1 + a_4 + a_7 + a_{10}) \omega + (a_2 + a_5 + a_8) \omega^2 = 0 \) (Equation 2)
- \( a_0 + a_3 + a_6 + a_9 + (a_2 + a_5 + a_8) \omega + (a_1 + a_4 + a_7 + a_{10}) \omega^2 = 0 \) (Equation 3)

5. **Solving the Equations**:
From Equations 2 and 3, we can isolate \( a_1 + a_4 + a_7 + a_{10} \) and \( a_2 + a_5 + a_8 \) and set up a system to solve for these sums.

Adding Equations 2 and 3 gives:
\[ 2(a_0 + a_3 + a_6 + a_9) + (a_1 + a_4 + a_7 + a_{10} + a_2 + a_5 + a_8)(\omega + \omega^2) = 0 \]
Since \( \omega + \omega^2 = -1 \):
\[ 2(a_0 + a_3 + a_6 + a_9) - (a_1 + a_4 + a_7 + a_{10} + a_2 + a_5 + a_8) = 0 \]

6. **Final Calculation**:
We can express \( a_1 + a_4 + a_7 + a_{10} \) in terms of \( a_0 + a_3 + a_6 + a_9 \):
\[ a_1 + a_4 + a_7 + a_{10} = 243 - (a_0 + a_3 + a_6 + a_9) \]
Since \( a_0 + a_3 + a_6 + a_9 \) can be calculated from the previous equations, we find:
\[ a_1 + a_4 + a_7 + a_{10} = \frac{243}{3} = 81 \]

Thus, the value of \( a_1 + a_4 + a_7 + a_{10} \) is **81**.