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 |[""^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)]|

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

|{:(.^(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

""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

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

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

sum_(r = 0)^(n-1) (C_r)/(C_r + C_(r+1)) =

For ""^(n) C_(r) + 2 ""^(n) C_(r-1) + ""^(n) C_(r-2) =

""^(n-2)C_(r)+2""^(n-2)C_(r-1)+""^(n-2)C_(r-2) equals :