If `Y=SX`, `Z=tX` all the variables being differentiable functions of `x` and lower suffices denote the derivative with respect to `x` and `|{:(X,Y,Z),(X_(1),Y_(1),Z_(1)),(X_(2),Y_(2),Z_(2)):}|+|{:(S_(1),t_(1)),(S_(2),t_(2)):}|=X^(n)`, then `n=`

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To solve the given problem step by step, we will analyze the determinants and apply the necessary operations to simplify them.

### Step 1: Write the Determinants
We start with the determinants given in the problem:
\[ \Delta_1 = \begin{vmatrix} X & Y & Z \\ X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \end{vmatrix} \]
and
\[ \Delta_2 = \begin{vmatrix} S_1 & t_1 \\ S_2 & t_2 \end{vmatrix} \]
We need to find the sum of these determinants and set it equal to \(X^n\).

### Step 2: Substitute for Y and Z
Given that \(Y = SX\) and \(Z = tX\), we can substitute these into the determinant:
\[ \Delta_1 = \begin{vmatrix} X & SX & tX \\ X_1 & S_1X_1 & t_1X_1 \\ X_2 & S_2X_2 & t_2X_2 \end{vmatrix} \]

### Step 3: Factor Out X
We can factor out \(X\) from each column:
\[ \Delta_1 = X \begin{vmatrix} 1 & S & t \\ X_1 & S_1X_1 & t_1X_1 \\ X_2 & S_2X_2 & t_2X_2 \end{vmatrix} \]

### Step 4: Compute the Derivatives
Next, we compute the derivatives:
- For \(Y\):
\[ Y_1 = S X_1 + S_1 X \]
\[ Y_2 = S X_2 + 2 S_1 X_1 + S_2 X \]
- For \(Z\):
\[ Z_1 = t X_1 + t_1 X \]
\[ Z_2 = t X_2 + 2 t_1 X_1 + t_2 X \]

### Step 5: Substitute Derivatives into the Determinant
Now we substitute these derivatives back into the determinant:
\[ \Delta_1 = \begin{vmatrix} X & SX & tX \\ X_1 & S_1X_1 + S X_1 & t_1X_1 + tX_1 \\ X_2 & S_2X_2 + 2 S_1X_1 & t_2X_2 + 2 t_1X_1 \end{vmatrix} \]

### Step 6: Apply Column Operations
We perform column operations:
- Column 2: \(C_2 - S C_1\)
- Column 3: \(C_3 - t C_1\)

This results in:
\[ \Delta_1 = X \begin{vmatrix} 1 & 0 & 0 \\ X_1 & S_1X_1 & t_1X_1 \\ X_2 & S_2X_2 & t_2X_2 \end{vmatrix} \]

### Step 7: Simplify the Determinant
Now we can simplify the determinant:
\[ \Delta_1 = X^2 S_1 T_1 \begin{vmatrix} 1 & 0 \\ X_2 & S_2 \end{vmatrix} \]

### Step 8: Final Determinant Calculation
The final determinant simplifies to:
\[ \Delta_1 = X^3 S_1 T_1 S_2 T_2 \]

### Step 9: Set Equal to \(X^n\)
We know from the problem statement that:
\[ \Delta_1 + \Delta_2 = X^n \]
Since \(\Delta_2\) is a smaller determinant, we can assume it contributes a lower degree term. Thus, we can equate:
\[ X^3 S_1 T_1 S_2 T_2 = X^n \]

### Step 10: Compare Powers
From the equation, we see that \(n = 3\).

### Final Answer
Thus, the value of \(n\) is:
\[ \boxed{3} \]