Evaluate `|{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(x)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|`

Answer

`(1)/(12) xyz (x-y) (y-z) (z-y)`

Explanation

`Delta=|{:(x,,(1)/(2)x(x-1),,(1)/(6)x(x-1)(x-2)),(y,,(1)/(2)y(y-1),,(1)/(6)y(y-1)(y-2)),(z,,(1)/(2)z(z-1),,(1)/(6)z(z-1)(z-2)):}|`
`=(1)/(12)xyz = |{:(1,,x-1,,x^(2)-3x+2),(1,,y-1,,y^(2)-3y+2),(1,,z-1,,z^(2)-3z+2):}|`
`=(1)/(12)xyz = |{:(1,,x,,x^(2)-3x+2),(1,,y,,y^(2)-3y+2),(1,,z,,z^(2)-3z+2):}|" "(C_(2) to C_(2) +C_(1))`
`=(1)/(12)xyz =|{:(1,,x,,x^(2)),(1,,y,,y^(2)),(1,,z,,z^(2)):}|" "(C_(3) to C_(3)+3C_(2)-2C_(1))`
`=(1)/(12)xyz (x-y) (y-z)(z-x)`