Which term in the expansion of `(2-3x)^(19)` has algebrically the last coefficients ? a. `10^(th)` b. `11^(th)` c. `12^(th)` d. `13^(th)`

Options

A. `10^(th)`
B. `11^(th)`
C. `12^(th)`
D. `13^(th)`

Explanation

To find which term in the expansion of \((2 - 3x)^{19}\) has the algebraically least coefficient, we will follow these steps:

### Step 1: Identify the general term in the binomial expansion.
The general term \(T_r\) in the expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r-1} a^{n-(r-1)} b^{r-1} \]
For our case, \(a = 2\), \(b = -3x\), and \(n = 19\). Thus, the general term becomes:
\[ T_r = \binom{19}{r-1} (2)^{19-(r-1)} (-3x)^{r-1} \]
This simplifies to:
\[ T_r = \binom{19}{r-1} (2)^{20-r} (-3)^{r-1} x^{r-1} \]

### Step 2: Find the coefficient of \(x^{r-1}\).
The coefficient of \(x^{r-1}\) in the term \(T_r\) is:
\[ \text{Coefficient} = \binom{19}{r-1} (2)^{20-r} (-3)^{r-1} \]

### Step 3: Determine the conditions for the least algebraic coefficient.
To find the term with the least algebraic coefficient, we need to consider the absolute values of the coefficients. We will calculate the absolute value of the coefficient:
\[ \left| \text{Coefficient} \right| = \binom{19}{r-1} (2)^{20-r} 3^{r-1} \]

### Step 4: Use the ratio of successive terms to find the maximum.
To find the term with the largest coefficient, we can use the ratio of successive terms:
\[ \frac{T_r}{T_{r-1}} = \frac{\binom{19}{r-1} (2)^{20-r} (-3)^{r-1}}{\binom{19}{r-2} (2)^{20-(r-1)} (-3)^{r-2}} \]
This simplifies to:
\[ \frac{T_r}{T_{r-1}} = \frac{19 - (r-1)}{r-1} \cdot \frac{2}{-3} \]
Setting this ratio equal to 1 to find the maximum term:
\[ \frac{19 - (r-1)}{r-1} \cdot \frac{2}{-3} = 1 \]

### Step 5: Solve for \(r\).
Rearranging gives:
\[ (19 - r + 1) \cdot 2 = -3(r - 1) \]
\[ (20 - r) \cdot 2 = -3r + 3 \]
\[ 40 - 2r = -3r + 3 \]
\[ 40 + 3 = 2r - r \]
\[ 43 = r \]
\[ r = 12 \]

### Step 6: Identify the term.
Thus, the term with the algebraically least coefficient corresponds to \(r = 12\), which means the \(12^{th}\) term has the least algebraic coefficient.

### Conclusion:
The term in the expansion of \((2 - 3x)^{19}\) that has the algebraically least coefficient is the \(12^{th}\) term.

### Final Answer:
**c. \(12^{th}\)**