The ratio of the coefficient of `x^(15)` to the term independent of x in the expansion of `(X^(2)+(2)/(x))^(15)` is

Options

A. `1:8`
B. `1:12`
C. `1:16`
D. `1:32`

Answer

To solve the problem, we need to find the ratio of the coefficient of \( x^{15} \) to the term independent of \( x \) in the expansion of \( (x^2 + \frac{2}{x})^{15} \).

### Step 1: Write the General Term
The general term in the expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
For our expression, we have \( a = x^2 \) and \( b = \frac{2}{x} \), and \( n = 15 \). Thus, the general term becomes:
\[ T_r = \binom{15}{r} (x^2)^{15-r} \left(\frac{2}{x}\right)^r \]

### Step 2: Simplify the General Term
Now, we simplify the general term:
\[ T_r = \binom{15}{r} (x^{2(15-r)}) \left(\frac{2^r}{x^r}\right) = \binom{15}{r} 2^r x^{30 - 3r} \]

### Step 3: Find the Coefficient of \( x^{15} \)
To find the coefficient of \( x^{15} \), we set the exponent of \( x \) to 15:
\[ 30 - 3r = 15 \]
Solving for \( r \):
\[ 3r = 30 - 15 \implies 3r = 15 \implies r = 5 \]
Now, we substitute \( r = 5 \) into the general term to find the coefficient:
\[ \text{Coefficient of } x^{15} = \binom{15}{5} 2^5 \]

### Step 4: Find the Coefficient of the Term Independent of \( x \)
To find the term independent of \( x \), we set the exponent of \( x \) to 0:
\[ 30 - 3r = 0 \]
Solving for \( r \):
\[ 3r = 30 \implies r = 10 \]
Now, we substitute \( r = 10 \) into the general term to find the coefficient:
\[ \text{Coefficient of the term independent of } x = \binom{15}{10} 2^{10} \]

### Step 5: Calculate the Required Ratio
Now, we need to find the ratio of the coefficient of \( x^{15} \) to the coefficient of the term independent of \( x \):
\[ \text{Required Ratio} = \frac{\binom{15}{5} 2^5}{\binom{15}{10} 2^{10}} \]
Since \( \binom{15}{10} = \binom{15}{5} \), we can simplify:
\[ \text{Required Ratio} = \frac{2^5}{2^{10}} = \frac{1}{2^5} = \frac{1}{32} \]

### Final Answer
Thus, the ratio of the coefficient of \( x^{15} \) to the term independent of \( x \) is:
\[ \frac{1}{32} \quad \text{or} \quad 1 : 32 \]