If `x^(4)` occurs in the rth term in the expansion of `( x^(4) + ( 1)/( x^(3)))^(15)` , then what is the value of r ?

Options

A. 7
B. 8
C. 9
D. 10

Answer

To find the value of \( r \) such that \( x^4 \) occurs in the \( r \)th term of the expansion of \( (x^4 + \frac{1}{x^3})^{15} \), we can follow these steps:

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = x^4 \), \( b = \frac{1}{x^3} \), and \( n = 15 \).

### Step 2: Write the General Term for Our Expansion
For our specific case:
\[ T_{r+1} = \binom{15}{r} (x^4)^{15-r} \left(\frac{1}{x^3}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{15}{r} x^{4(15-r)} \cdot \frac{1}{x^{3r}} = \binom{15}{r} x^{60 - 4r - 3r} = \binom{15}{r} x^{60 - 7r} \]

### Step 3: Set the Exponent Equal to 4
We want the term to contain \( x^4 \):
\[ 60 - 7r = 4 \]

### Step 4: Solve for \( r \)
Rearranging the equation gives:
\[ 60 - 4 = 7r \\ 56 = 7r \\ r = \frac{56}{7} = 8 \]

### Conclusion
Thus, the value of \( r \) is \( 8 \).