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+1)|=|^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+1)|` .

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)| .

Show that |[""^xC_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)]|

Without expanding show that : |(""^xC_r,""^xC_(r+1),""^xC_(r+2)),(""^yC_r,""^yC_(r+1),""^yC_(r+2)),(""^zC_r,""^zC_(r+1),""^zC_(r+2))|=|(""^xC_r,""^(x+1)C_(r+1),""^(x+2)C_(r+2)),(""^yC_r,""^(y+1)C_(r+1),""^(y+2)C_(r+2)),(""^zC_r,""^(z+1)C_(r+1),""^(z+2)C_(r+2))|

|(""^(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+1)C_(r+2))| a)0 b) 2^(n) c) x+y+z_(C_(r)) d) x+y+""^(z)C_(r)

|{:(.^(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)):}| is equal to

|{:(.^(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)):}| is equal to

|{:(.^(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)):}| is equal to

Show that .^(n)C_(r)+.^(n-1)C_(r-1)+.^(n-1)C_(r-2)=.^(n+1)C_(r) .