In the expansion of `(1+ x + 7/x)^11`find the term not containing x.

Explanation

To find the term not containing \( x \) in the expansion of \( (1 + x + \frac{7}{x})^{11} \), we can follow these steps:

### Step 1: Identify the general term in the expansion
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, let \( a = 1 \), \( b = x \), and \( c = \frac{7}{x} \). Therefore, the general term becomes:
\[ T = \frac{11!}{r_1! r_2! r_3!} (1)^{r_1} (x)^{r_2} \left(\frac{7}{x}\right)^{r_3} \]
This simplifies to:
\[ T = \frac{11!}{r_1! r_2! r_3!} \cdot 7^{r_3} \cdot x^{r_2 - r_3} \]

### Step 2: Set the exponent of \( x \) to zero
To find the term that does not contain \( x \), we need the exponent of \( x \) to be zero:
\[ r_2 - r_3 = 0 \implies r_2 = r_3 \]

### Step 3: Express \( r_1 \) in terms of \( r_2 \)
Since \( r_1 + r_2 + r_3 = 11 \) and \( r_2 = r_3 \), we can substitute \( r_3 \) with \( r_2 \):
\[ r_1 + r_2 + r_2 = 11 \implies r_1 + 2r_2 = 11 \implies r_1 = 11 - 2r_2 \]

### Step 4: Determine possible values of \( r_2 \)
Since \( r_1 \) must be non-negative, we have:
\[ 11 - 2r_2 \geq 0 \implies r_2 \leq \frac{11}{2} = 5.5 \]
Thus, \( r_2 \) can take integer values from \( 0 \) to \( 5 \).

### Step 5: Calculate the terms for \( r_2 = r_3 \)
Now, we will find the terms for \( r_2 = 0, 1, 2, 3, 4, 5 \):

1. **For \( r_2 = 0 \)**:
- \( r_1 = 11 \), \( r_3 = 0 \)
- \( T_1 = \frac{11!}{11!0!0!} \cdot 7^0 = 1 \)

2. **For \( r_2 = 1 \)**:
- \( r_1 = 9 \), \( r_3 = 1 \)
- \( T_2 = \frac{11!}{9!1!1!} \cdot 7^1 = \frac{11 \times 10}{2} \cdot 7 = 55 \cdot 7 = 385 \)

3. **For \( r_2 = 2 \)**:
- \( r_1 = 7 \), \( r_3 = 2 \)
- \( T_3 = \frac{11!}{7!2!2!} \cdot 7^2 = \frac{11 \times 10 \times 9 \times 8}{4} \cdot 49 = 495 \cdot 49 = 24255 \)

4. **For \( r_2 = 3 \)**:
- \( r_1 = 5 \), \( r_3 = 3 \)
- \( T_4 = \frac{11!}{5!3!3!} \cdot 7^3 = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6}{6} \cdot 343 = 27720 \cdot 343 = 9502440 \)

5. **For \( r_2 = 4 \)**:
- \( r_1 = 3 \), \( r_3 = 4 \)
- \( T_5 = \frac{11!}{3!4!4!} \cdot 7^4 = \frac{11 \times 10 \times 9 \times 8}{24} \cdot 2401 = 330 \cdot 2401 = 792330 \)

6. **For \( r_2 = 5 \)**:
- \( r_1 = 1 \), \( r_3 = 5 \)
- \( T_6 = \frac{11!}{1!5!5!} \cdot 7^5 = \frac{11 \times 10}{120} \cdot 16807 = 9.1667 \cdot 16807 = 154970 \)

### Step 6: Sum the coefficients
Now, we sum all the coefficients of the terms that do not contain \( x \):
\[ 1 + 385 + 24255 + 9502440 + 792330 + 154970 = 10381781 \]

### Final Answer
The term not containing \( x \) in the expansion of \( (1 + x + \frac{7}{x})^{11} \) is \( 10381781 \).

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