The value of n for which the determinant `Delta=|(""^(8)C_3,""^(9)C_5,""^(10)C_7),(""^(8)C_4,""^(9)C_6,""^(10)C_8),(""^(9)C_n,""^(10)C_(n+2),""^(11)C_(n+4))|` = 0 is

Options

A. 2
B. 3
C. 4
D. none

Answer

To find the value of \( n \) for which the determinant

\[ \Delta = \begin{vmatrix} {^{8}C_{3}} & {^{9}C_{5}} & {^{10}C_{7}} \\ {^{8}C_{4}} & {^{9}C_{6}} & {^{10}C_{8}} \\ {^{9}C_{n}} & {^{10}C_{n+2}} & {^{11}C_{n+4}} \end{vmatrix} = 0, \]

we will follow these steps:

### Step 1: Use the properties of combinations
Recall the identity for combinations:
\[ {^nC_r} + {^nC_{r+1}} = {^{n+1}C_{r+1}}. \]
Using this identity, we can simplify the first two rows of the determinant.

### Step 2: Simplify the first two rows
We can add the first row to the second row:
\[ R_2 \rightarrow R_2 + R_1. \]
This gives us:
\[ \Delta = \begin{vmatrix} {^{8}C_{3}} & {^{9}C_{5}} & {^{10}C_{7}} \\ {^{8}C_{4} + ^{8}C_{3}} & {^{9}C_{6} + ^{9}C_{5}} & {^{10}C_{8} + ^{10}C_{7}} \\ {^{9}C_{n}} & {^{10}C_{n+2}} & {^{11}C_{n+4}} \end{vmatrix}. \]

### Step 3: Apply the combination identity
Using the combination identity, we have:
\[ {^{8}C_{4} + ^{8}C_{3}} = {^{9}C_{4}}, \quad {^{9}C_{6} + ^{9}C_{5}} = {^{10}C_{6}}, \quad {^{10}C_{8} + ^{10}C_{7}} = {^{11}C_{8}}. \]
Thus, the determinant becomes:
\[ \Delta = \begin{vmatrix} {^{8}C_{3}} & {^{9}C_{5}} & {^{10}C_{7}} \\ {^{9}C_{4}} & {^{10}C_{6}} & {^{11}C_{8}} \\ {^{9}C_{n}} & {^{10}C_{n+2}} & {^{11}C_{n+4}} \end{vmatrix}. \]

### Step 4: Set the determinant to zero
For the determinant to be equal to zero, the rows must be linearly dependent. This can happen if two rows are identical or proportional.

### Step 5: Analyze the rows
We can set the second row equal to the first row:
\[ {^{9}C_{4}} = {^{9}C_{n}}, \quad {^{10}C_{6}} = {^{10}C_{n+2}}, \quad {^{11}C_{8}} = {^{11}C_{n+4}}. \]

### Step 6: Solve for \( n \)
From the first equation:
\[ n = 4. \]
From the second equation:
\[ n + 2 = 6 \implies n = 4. \]
From the third equation:
\[ n + 4 = 8 \implies n = 4. \]

### Conclusion
All equations consistently lead to \( n = 4 \). Therefore, the value of \( n \) for which the determinant is zero is:

\[ \boxed{4}. \]