To solve the problem of finding the term independent of \( x \) in the expression \( (1 + x + x^{-2} + x^{-3})^{10} \) and then determining the last digit of \( (n + 2)^n \), where \( n \) is the coefficient of that term, we can follow these steps:
### Step 1: Identify the expression
We start with the expression:
\[
(1 + x + x^{-2} + x^{-3})^{10}
\]
### Step 2: Rewrite the expression
We can rewrite the terms involving negative powers of \( x \):
\[
1 + x + \frac{1}{x^2} + \frac{1}{x^3} = 1 + x + x^{-2} + x^{-3}
\]
This can be rewritten as:
\[
1 + x + \frac{1 + x^3}{x^2}
\]
### Step 3: Find the constant term
To find the term independent of \( x \), we need to consider the expansion of the expression. The term independent of \( x \) occurs when the powers of \( x \) sum to zero.
### Step 4: Use the multinomial expansion
Using the multinomial theorem, we can express the expansion:
\[
(1 + x + x^{-2} + x^{-3})^{10}
\]
We need to find combinations of the terms that yield a total power of \( x^0 \).
### Step 5: Set up the equation
Let \( a \), \( b \), \( c \), and \( d \) be the number of times we choose \( 1 \), \( x \), \( x^{-2} \), and \( x^{-3} \) respectively. We have:
\[
a + b + c + d = 10
\]
The power of \( x \) in the term is given by:
\[
b - 2c - 3d = 0
\]
### Step 6: Solve the equations
From \( b - 2c - 3d = 0 \), we can express \( b \) in terms of \( c \) and \( d \):
\[
b = 2c + 3d
\]
Substituting this into the first equation gives:
\[
a + (2c + 3d) + c + d = 10
\]
This simplifies to:
\[
a + 3c + 4d = 10
\]
### Step 7: Find integer solutions
We need to find non-negative integer solutions for \( a + 3c + 4d = 10 \). We can try different values for \( d \) and solve for \( a \) and \( c \).
1. **If \( d = 0 \)**:
\[
a + 3c = 10 \quad \Rightarrow \quad (10, 0), (7, 1), (4, 2), (1, 3)
\]
This gives us 4 solutions.
2. **If \( d = 1 \)**:
\[
a + 3c = 6 \quad \Rightarrow \quad (6, 0), (3, 1), (0, 2)
\]
This gives us 3 solutions.
3. **If \( d = 2 \)**:
\[
a + 3c = 2 \quad \Rightarrow \quad (2, 0)
\]
This gives us 1 solution.
4. **If \( d \geq 3 \)**:
No solutions since \( 4d > 10 \).
### Step 8: Count the total solutions
Adding the solutions:
- For \( d = 0 \): 4 solutions
- For \( d = 1 \): 3 solutions
- For \( d = 2 \): 1 solution
Total solutions = \( 4 + 3 + 1 = 8 \).
### Step 9: Calculate \( n \)
Thus, the coefficient \( n \) of the term independent of \( x \) is \( 8 \).
### Step 10: Find the last digit of \( (n + 2)^n \)
Now we need to calculate:
\[
(n + 2)^n = (8 + 2)^8 = 10^8
\]
The last digit of \( 10^8 \) is \( 0 \).
### Final Answer
The last digit of \( (n + 2)^n \) is \( \boxed{0} \).
