If `n gt 3`, then `xyz^(n)C_(0)-(x-1)(y-1)(z-1)""^(n)C_(1)+(x-2)(y-2)(z-2)""^(n)C_(2)-`
`(x-3)(y-3)(z-3)""^(n)C_(3)+…..+(-1)^(n)(x-n)(y-n)(z-n)""^(n)C_(n)` equals :

Options

A. `xy xx2^(n)`
B. n xy
C. xy
D. none of these

Answer

d

Explanation

We have ,
`xy C_(0) - (x -1) (y -1)C_(1) +(x -2) (y -2)C_(2)- (x-3) (y -3) C_(3)`
`+ …+ (-1)^(n) (x -n) (y -n) C_(n)`
`= sum_(r=0)^(n) (-1)^(r) (x -r) (y -r) C_(r)`
`= sum_(r=0)^(n) (-1)^(r){xy - (x+y)r + r^(2)}C_(r)`
=`sum_(r=0)^(n) (-1)^(r)xyC_(r) -sum_(r=0)^(n) (-1)^(r) r (x+y)C_(r) +sum_(r=0)^(n)(-1)^(r) r^(2)C_(r)`
=` xy sum_(r=0)^(n) (-1)^(r)C_(r)(x -y) sum_(r=0)^(n) (-1)^(r) rC_(r) +sum_(r=0)^(n)(-1)^(r) r^(2)C_(r)`
`= xy xx0 - (x + y )xx 0 + 0 = 0`
`[because sum_(r=0)^(n) (-1)^(r)C_(r)= sum_(r=0)^(n) (-1)^(r) rC_(r) =sum_(r=0)^(n)(-1)^(r) r^(2)C_(r) = 0]`