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 value of \( a_1 + a_4 + a_7 + a_{10} \) from the expansion of \( (1 + x + x^2)^5 \).

### Step-by-Step Solution:

1. **Identify the Expression**: We start with the expression \( (1 + x + x^2)^5 \). We want to expand this and find the coefficients \( a_1, a_4, a_7, \) and \( a_{10} \).

2. **Use Roots of Unity**: To find the coefficients efficiently, we can use the roots of unity. Let \( \omega = e^{2\pi i / 3} \), which is a primitive cube root of unity. The property of cube roots of unity is that \( 1 + \omega + \omega^2 = 0 \).

3. **Evaluate at \( x = 1 \)**: First, we substitute \( x = 1 \):
\[ (1 + 1 + 1^2)^5 = 3^5 = 243 \]
This gives us:
\[ a_0 + a_1 + a_2 + \ldots + a_{10} = 243 \]

4. **Evaluate at \( x = \omega \)**: Next, we substitute \( x = \omega \):
\[ (1 + \omega + \omega^2)^5 = 0^5 = 0 \]
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:
\[ a_0 + a_1 \omega + a_2 \omega^2 + a_3 + a_4 \omega + a_5 \omega^2 + a_6 + a_7 \omega + a_8 \omega^2 + a_9 + a_{10} \omega = 0 \]

5. **Evaluate at \( x = \omega^2 \)**: Now, we substitute \( x = \omega^2 \):
\[ (1 + \omega^2 + (\omega^2)^2)^5 = 0^5 = 0 \]
This gives us:
\[ a_0 + a_1 \omega^2 + a_2 \omega + a_3 + a_4 \omega^2 + a_5 \omega + a_6 + a_7 \omega^2 + a_8 \omega + a_9 + a_{10} \omega^2 = 0 \]

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

7. **Combine Equations**: By adding equations (2) and (3), we can isolate terms involving \( a_1, a_4, a_7, \) and \( a_{10} \).

8. **Solve for the Coefficients**: From the symmetry and properties of roots of unity, we find that:
\[ 3(a_1 + a_4 + a_7 + a_{10}) = 243 \]
Thus,
\[ a_1 + a_4 + a_7 + a_{10} = \frac{243}{3} = 81 \]

### Final Answer:
The value of \( a_1 + a_4 + a_7 + a_{10} \) is \( \boxed{81} \).