The constant term in the expansion of `(2x+1/x^7+3x^2)^5` is___.

Answer

To find the constant term in the expansion of \((2x + \frac{1}{x^7} + 3x^2)^5\), we can use the multinomial expansion. Let's break down the solution step by step.

### Step 1: Identify the General Term
The general term in the expansion of \((a + b + c)^n\) is given by:
\[ T = \frac{n!}{r_1! r_2! r_3!} a^{r_1} b^{r_2} c^{r_3} \]
where \(r_1 + r_2 + r_3 = n\).

In our case:
- \(a = 2x\)
- \(b = \frac{1}{x^7}\)
- \(c = 3x^2\)
- \(n = 5\)

Thus, the general term \(T\) can be written as:
\[ T = \frac{5!}{r_1! r_2! r_3!} (2x)^{r_1} \left(\frac{1}{x^7}\right)^{r_2} (3x^2)^{r_3} \]

### Step 2: Simplify the General Term
Substituting the values into the general term:
\[ T = \frac{5!}{r_1! r_2! r_3!} (2^{r_1} x^{r_1}) \left(\frac{1}{x^{7r_2}}\right) (3^{r_3} x^{2r_3}) \]
This simplifies to:
\[ T = \frac{5!}{r_1! r_2! r_3!} 2^{r_1} 3^{r_3} x^{r_1 - 7r_2 + 2r_3} \]

### Step 3: Find the Condition for the Constant Term
For \(T\) to be a constant term, the exponent of \(x\) must be zero:
\[ r_1 - 7r_2 + 2r_3 = 0 \]

### Step 4: Set Up the Equations
We also have the equation from the multinomial expansion:
\[ r_1 + r_2 + r_3 = 5 \]

Now we have two equations:
1. \(r_1 - 7r_2 + 2r_3 = 0\)
2. \(r_1 + r_2 + r_3 = 5\)

### Step 5: Solve the Equations
From the second equation, we can express \(r_1\) in terms of \(r_2\) and \(r_3\):
\[ r_1 = 5 - r_2 - r_3 \]

Substituting \(r_1\) into the first equation:
\[ (5 - r_2 - r_3) - 7r_2 + 2r_3 = 0 \]
This simplifies to:
\[ 5 - 8r_2 + r_3 = 0 \implies r_3 = 8r_2 - 5 \]

### Step 6: Substitute Back
Now substitute \(r_3\) back into the second equation:
\[ r_1 + r_2 + (8r_2 - 5) = 5 \]
This simplifies to:
\[ r_1 + 9r_2 - 5 = 5 \implies r_1 + 9r_2 = 10 \implies r_1 = 10 - 9r_2 \]

### Step 7: Find Valid Values
Since \(r_1\), \(r_2\), and \(r_3\) must be non-negative integers, we can find valid values for \(r_2\):
- If \(r_2 = 0\), then \(r_1 = 10\) (not valid since \(r_1 + r_2 + r_3 = 5\))
- If \(r_2 = 1\), then \(r_1 = 1\) and \(r_3 = 3\) (valid)
- If \(r_2 = 2\), then \(r_1 = -8\) (not valid)

Thus, the only valid solution is:
- \(r_1 = 1\)
- \(r_2 = 1\)
- \(r_3 = 3\)

### Step 8: Calculate the Constant Term
Now substituting \(r_1\), \(r_2\), and \(r_3\) into the general term:
\[ T = \frac{5!}{1!1!3!} (2^1)(3^3) \]
Calculating this gives:
\[ T = \frac{120}{1 \cdot 1 \cdot 6} \cdot 2 \cdot 27 = 20 \cdot 54 = 1080 \]

### Final Answer
The constant term in the expansion is:
\[ \boxed{1080} \]