If ` (1 +x + x^(2) + …+ x^(9))^(4) (x + x^(2) + x^(3) + … + x^(9))`
` = sum_(r=1)^(45) a_(r) x^(r)` and the value of ` a_(2) + a_(6) + a_(10) + … + a_(42) " is " lambda `
the sum of all digits of ` lambda ` is .

Answer

To solve the given problem step by step, we will analyze the expression and compute the required values systematically.

### Step 1: Simplify the Expression
The expression given is:
\[ (1 + x + x^2 + \ldots + x^9)^4 (x + x^2 + x^3 + \ldots + x^9) \]
The first part, \(1 + x + x^2 + \ldots + x^9\), is a geometric series. The sum of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Here, \(a = 1\), \(r = x\), and \(n = 10\):
\[ 1 + x + x^2 + \ldots + x^9 = \frac{1 - x^{10}}{1 - x} \]
Thus, we have:
\[ (1 + x + x^2 + \ldots + x^9)^4 = \left(\frac{1 - x^{10}}{1 - x}\right)^4 \]

The second part, \(x + x^2 + x^3 + \ldots + x^9\), can also be simplified:
\[ x + x^2 + x^3 + \ldots + x^9 = x(1 + x + x^2 + \ldots + x^8) = x \cdot \frac{1 - x^9}{1 - x} = \frac{x(1 - x^9)}{1 - x} \]

### Step 2: Combine the Expressions
Now, combining both parts, we get:
\[ \left(\frac{1 - x^{10}}{1 - x}\right)^4 \cdot \frac{x(1 - x^9)}{1 - x} = \frac{x(1 - x^{10})^4(1 - x^9)}{(1 - x)^5} \]

### Step 3: Find Coefficients
We know that this expression can be represented as:
\[ \sum_{r=1}^{45} a_r x^r \]
We need to find the sum \(a_2 + a_6 + a_{10} + \ldots + a_{42}\).

### Step 4: Evaluate at Specific Values
To find the coefficients, we will evaluate the expression at specific values of \(x\).

1. **Evaluate at \(x = 1\)**:
\[ (1 + 1 + 1 + \ldots + 1)^4 \cdot (1 + 1 + 1 + \ldots + 1) = 10^4 \cdot 9 = 90000 \]
This gives us:
\[ a_1 + a_2 + a_3 + \ldots + a_{45} = 90000 \quad \text{(Equation 1)} \]

2. **Evaluate at \(x = -1\)**:
\[ (1 - 1 + 1 - 1 + \ldots)^4 \cdot (-1 + 1 - 1 + \ldots) = 0 \]
This gives us:
\[ -a_1 + a_2 - a_3 + a_4 - \ldots - a_{45} = 0 \quad \text{(Equation 2)} \]

3. **Evaluate at \(x = i\)**:
The sum of four consecutive powers of \(i\) is zero. Thus:
\[ (1 + i + i^2 + i^3)^4 \cdot (i + i^2 + i^3 + i^4) = 0 \]

4. **Evaluate at \(x = -i\)**:
Similarly, we will have:
\[ -4i = a_1 i + a_2 - a_3 i - a_4 + \ldots \quad \text{(Equation 4)} \]

### Step 5: Combine Equations
Adding Equations 1, 2, 3, and 4 gives:
\[ 4(a_2 + a_6 + a_{10} + \ldots + a_{42}) = 90000 \]
Thus:
\[ a_2 + a_6 + a_{10} + \ldots + a_{42} = \frac{90000}{4} = 22500 \]
Let \(\lambda = 22500\).

### Step 6: Calculate the Sum of Digits
Now, we need to find the sum of the digits of \(\lambda = 22500\):
\[ 2 + 2 + 5 + 0 + 0 = 9 \]

### Final Answer
The sum of all digits of \(\lambda\) is:
\[ \boxed{9} \]