Co-efficient of `alpha^t` in thet expansion of `(alpha+p)^(m-1)+(alpha+p)^(m-2)(alpha+q)+(alpha+p)^(m-3)(alpha+q)^2+dot(alpha+q)^(m-1)` where `alpha != -q and p !=q` is :

Options

A. `(.^(m)C_(t)(p^(t) - q^(t)))/(p-q)`
B. `(.^(m)C_(t)(p^(m-t) - q^(m-t)))/(p-q)`
C. `(.^(m)C_(t)(p^(t) - q^(t)))/(p-q)`
D. `(.^(m)C_(t)(p^(m-t) + q^(m-t)))/(p-q)`

Answer

To find the coefficient of \( \alpha^t \) in the expansion of

\[ S = ( \alpha + p)^{m-1} + ( \alpha + p)^{m-2} ( \alpha + q) + ( \alpha + p)^{m-3} ( \alpha + q)^2 + \ldots + ( \alpha + q)^{m-1}, \]

we can follow these steps:

### Step 1: Identify the series
We can rewrite the series \( S \) as a sum of terms where each term is of the form \( ( \alpha + p)^{m-k} ( \alpha + q)^{k} \) for \( k = 0, 1, 2, \ldots, m-1 \).

### Step 2: Express \( S \) as a geometric series
The series can be recognized as a geometric series with:
- First term \( A = ( \alpha + p)^{m-1} \)
- Common ratio \( R = \frac{ \alpha + q}{ \alpha + p} \)
- Number of terms \( n = m \)

### Step 3: Use the formula for the sum of a geometric series
The sum of a geometric series is given by:

\[ S_n = A \frac{1 - R^n}{1 - R} \]

Substituting our values, we have:

\[ S = ( \alpha + p)^{m-1} \frac{1 - \left( \frac{ \alpha + q}{ \alpha + p} \right)^m}{1 - \frac{ \alpha + q}{ \alpha + p}} \]

### Step 4: Simplify the expression
We simplify the denominator:

\[ 1 - \frac{ \alpha + q}{ \alpha + p} = \frac{( \alpha + p) - ( \alpha + q)}{ \alpha + p} = \frac{p - q}{ \alpha + p} \]

Thus, we can rewrite \( S \):

\[ S = ( \alpha + p)^{m-1} \cdot \frac{( \alpha + p)^m - ( \alpha + q)^m}{p - q} \]

### Step 5: Expand using the binomial theorem
Now, we need to find the coefficient of \( \alpha^t \) in the expression:

\[ S = \frac{( \alpha + p)^{m} - ( \alpha + q)^{m}}{p - q} \]

Using the binomial theorem, we can expand both \( ( \alpha + p)^{m} \) and \( ( \alpha + q)^{m} \):

\[ ( \alpha + p)^{m} = \sum_{t=0}^{m} \binom{m}{t} \alpha^t p^{m-t} \]
\[ ( \alpha + q)^{m} = \sum_{t=0}^{m} \binom{m}{t} \alpha^t q^{m-t} \]

### Step 6: Combine the expansions
The coefficient of \( \alpha^t \) in \( S \) can be found by subtracting the coefficients from the two expansions:

\[ \text{Coefficient of } \alpha^t = \frac{1}{p - q} \left( \binom{m}{t} p^{m-t} - \binom{m}{t} q^{m-t} \right) \]

### Step 7: Final expression
Thus, we have:

\[ \text{Coefficient of } \alpha^t = \frac{\binom{m}{t} (p^{m-t} - q^{m-t})}{p - q} \]

### Conclusion
The coefficient of \( \alpha^t \) in the given expansion is:

\[ \frac{\binom{m}{t} (p^{m-t} - q^{m-t})}{p - q} \]