The coefficient of `x^(-17)` in `(x^4 - (1)/(x^3))^15` is

Options

A. `-1365`
B. 1365
C. 465
D. `-465`

Answer

To find the coefficient of \( x^{-17} \) in the expression \( (x^4 - \frac{1}{x^3})^{15} \), we can follow these steps:

### Step 1: Identify the general term
The expression can be expanded using the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

In our case, let \( a = x^4 \) and \( b = -\frac{1}{x^3} \), and \( n = 15 \). The general term \( T_{r+1} \) in this expansion is given by:

\[ T_{r+1} = \binom{15}{r} (x^4)^{15-r} \left(-\frac{1}{x^3}\right)^r \]

### Step 2: Simplify the general term
Now we simplify the general term:

\[ T_{r+1} = \binom{15}{r} (x^{4(15-r)}) \left(-1\right)^r \left(\frac{1}{x^{3r}}\right) \]

This can be rewritten as:

\[ T_{r+1} = \binom{15}{r} (-1)^r x^{60 - 4r - 3r} = \binom{15}{r} (-1)^r x^{60 - 7r} \]

### Step 3: Set the exponent equal to -17
We need to find the value of \( r \) such that the exponent of \( x \) equals -17:

\[ 60 - 7r = -17 \]

### Step 4: Solve for \( r \)
Rearranging the equation gives:

\[ 60 + 17 = 7r \implies 77 = 7r \implies r = \frac{77}{7} = 11 \]

### Step 5: Substitute \( r \) back into the general term
Now we substitute \( r = 11 \) back into the general term to find the coefficient:

\[ T_{12} = \binom{15}{11} (-1)^{11} x^{60 - 7 \cdot 11} \]

Calculating \( 60 - 77 = -17 \), we have:

\[ T_{12} = \binom{15}{11} (-1)^{11} x^{-17} \]

### Step 6: Calculate the binomial coefficient
The binomial coefficient \( \binom{15}{11} \) can be calculated as:

\[ \binom{15}{11} = \binom{15}{4} = \frac{15!}{4!(15-4)!} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = \frac{32760}{24} = 1365 \]

### Step 7: Determine the final coefficient
Thus, the coefficient of \( x^{-17} \) is:

\[ T_{12} = 1365 \cdot (-1) = -1365 \]

### Final Answer
The coefficient of \( x^{-17} \) in \( (x^4 - \frac{1}{x^3})^{15} \) is \( \boxed{-1365} \).

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