The coefficient of `t^4` in `((1-t^6)/(1-t))^3` (a) `18` (b) `12` (c) `9` (d) `15`

Options

A. 12
B. 15
C. 10
D. 14

Answer

To find the coefficient of \( t^4 \) in the expression \( \left( \frac{1 - t^6}{1 - t} \right)^3 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left( \frac{1 - t^6}{1 - t} \right)^3 = (1 - t^6)^3 \cdot (1 - t)^{-3} \]

### Step 2: Expand \( (1 - t^6)^3 \)
Using the binomial theorem, we can expand \( (1 - t^6)^3 \):
\[ (1 - t^6)^3 = \sum_{k=0}^{3} \binom{3}{k} (-1)^k (t^6)^k = 1 - 3t^6 + 3t^{12} - t^{18} \]

### Step 3: Expand \( (1 - t)^{-3} \)
Using the generalized binomial series, we can expand \( (1 - t)^{-3} \):
\[ (1 - t)^{-3} = \sum_{n=0}^{\infty} \binom{n + 2}{2} t^n \]
This series gives us the coefficients for \( t^n \) as \( \binom{n + 2}{2} \).

### Step 4: Find the coefficient of \( t^4 \)
We need to find the coefficient of \( t^4 \) in the product of the two expansions. We can identify the contributions to \( t^4 \) from the terms of \( (1 - t^6)^3 \):
- From \( 1 \): The coefficient of \( t^4 \) from \( (1 - t)^{-3} \) is \( \binom{4 + 2}{2} = \binom{6}{2} = 15 \).
- From \( -3t^6 \): The coefficient of \( t^{-2} \) from \( (1 - t)^{-3} \) is \( 0 \) (since there is no \( t^{-2} \) term).
- From \( 3t^{12} \) and \( -t^{18} \): Similarly, these terms do not contribute to \( t^4 \).

Thus, the only contribution to the coefficient of \( t^4 \) comes from the first term, which is \( 15 \).

### Final Answer
The coefficient of \( t^4 \) in \( \left( \frac{1 - t^6}{1 - t} \right)^3 \) is \( 15 \).