Show that `|^x C_r^x C_(r+1)^x C_(r+2)^y C_r^y C_(r+1)^y C_(r+2)^z C_r^z C_(r+1)^z C_(r+2)|=|^x C_r^(x+1)C_(r+1)^(x+2)C_(r+2)^y C_r^(y+1)C_(r+1)^(y+2)C_(r+2)^z C_r^(z+1)C_(r+1)^(z+2)C_(r+2)|` .

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)`