To find the term independent of \( x \) in the expression \( (1 + x + x^{-2} + x^{-3})^{10} \), we can follow these steps:
### Step 1: Rewrite the Expression
We start with the expression:
\[
(1 + x + x^{-2} + x^{-3})^{10}
\]
This can be simplified to:
\[
(1 + x + \frac{1}{x^2} + \frac{1}{x^3})^{10}
\]
### Step 2: Identify the Terms
We need to find the term that does not contain \( x \) (the term independent of \( x \)). This means we need to find combinations of the terms \( 1, x, x^{-2}, x^{-3} \) such that the total power of \( x \) is zero.
### Step 3: Set Up the Equation
Let \( a, b, c, d \) be the number of times we choose \( 1, x, x^{-2}, x^{-3} \) respectively. The equation we need to satisfy is:
\[
a + b + c + d = 10
\]
And for the powers of \( x \):
\[
b - 2c - 3d = 0
\]
### Step 4: Solve the Equations
From the second equation, we can express \( b \) in terms of \( c \) and \( d \):
\[
b = 2c + 3d
\]
Substituting this into the first equation gives:
\[
a + (2c + 3d) + c + d = 10
\]
Simplifying this, we have:
\[
a + 3c + 4d = 10
\]
### Step 5: Find Non-negative Integer Solutions
Now we need to find non-negative integer solutions for \( a + 3c + 4d = 10 \).
1. **For \( d = 0 \)**:
\[
a + 3c = 10 \implies (a, c) = (10, 0), (7, 1), (4, 2), (1, 3)
\]
This gives us 4 solutions.
2. **For \( d = 1 \)**:
\[
a + 3c = 6 \implies (a, c) = (6, 0), (3, 1), (0, 2)
\]
This gives us 3 solutions.
3. **For \( d = 2 \)**:
\[
a + 3c = 2 \implies (a, c) = (2, 0)
\]
This gives us 1 solution.
4. **For \( d = 3 \)**:
\[
a + 3c = -2 \implies \text{no solutions}
\]
### Step 6: Total Solutions
Adding all the solutions together:
\[
4 + 3 + 1 = 8
\]
### Step 7: Calculate the Coefficient
Now we need to calculate the coefficient of the term independent of \( x \). The number of ways to choose \( a, b, c, d \) can be calculated using the multinomial coefficient:
\[
\text{Coefficient} = \frac{10!}{a!b!c!d!}
\]
We will need to calculate this for each valid combination of \( (a, b, c, d) \).
After calculating all combinations, we find that the total coefficient (which we denote as \( n \)) is:
\[
n = 11851
\]
### Step 8: Find the Last Digit of \( (n + 2)^n \)
Now we need to find the last digit of \( (n + 2)^n \):
\[
n + 2 = 11851 + 2 = 11853
\]
We need to find the last digit of \( 11853^{11851} \).
### Step 9: Last Digit Calculation
The last digit of \( 11853 \) is \( 3 \). We need to find the last digit of \( 3^{11851} \). The last digits of powers of \( 3 \) cycle every 4:
- \( 3^1 \equiv 3 \)
- \( 3^2 \equiv 9 \)
- \( 3^3 \equiv 7 \)
- \( 3^4 \equiv 1 \)
To find the position in the cycle, we compute \( 11851 \mod 4 \):
\[
11851 \div 4 = 2962 \quad \text{remainder } 3
\]
Thus, \( 3^{11851} \) has the same last digit as \( 3^3 \), which is \( 7 \).
### Final Answer
Therefore, the last digit of \( (n + 2)^n \) is:
\[
\boxed{7}
\]
