the value of the determinant
`|{:(.^(n)C_(r-1),,.^(n)C_(r),,(r+1)^(n+2)C_(r+1)),(.^(n)C_(r),,.^(n)C_(r+1),,(r+2)^(n+2)C_(r+2)),(.^(n)C_(r+1),,.^(n)C_(r+2),,(r+3)^(n+2)C_(r+3)):}|` is

Options

A. `n^(2)+n-1)`
B. `0`
C. `.^(n+3)C_(r+3)`
D. `.^(n)C_(r-1)+^(n)C_(r)+^(n)C_(r+1)`

Answer

B

Explanation

`Delta = |{:(.^(n)C_(r-1),,.^(n)C_(r),,(r+1)^(n+2)C_(r+1)),(.^(n)C_(r),,.^(n)C_(r+1),,(n+2)^(n+1)C_(r+1)),(.^(n)C_(r+1),,.^(n)C_(r+2),,(r+3)^(n+2)C_(r+3)):}|`
Applying `C_(1) to C_(1)+C_(2) " and using " .^(n)C_(r)=(n)/(r )^(n-1) C_(r+1) " in " C_(3)` we get
` Delta =|{:(.^(n+1)C_(r),,.^(n)C_(r),,(n+2)^(n+1)C_(r)),(.^(n+1)C_(r+1),,.^(n)C_(r+1),,(n+2)^(n+1)C_(r+1)),(.^(n+1)C_(r+2),,.^(n)C_(r+2),,(n+2)^(n+1)C_(r+2)):}|`
`=(n+2) |{:(.^(n+1)C_(r),,.^(n)C_(r),,.^(n+1)C_(r)),(.^(n+1)C_(r+1),,.^(n)C_(r+1),,.^(n+1)C_(r+1)),(.^(n+1)C_(r+2),,.^(n)C_(r+2),,.^(n+1)C_(r+2)):}|`
( as `C_(1) " and " C_(3)`are identical )