Prove that : `.^(n-1)C_(r)+.^(n-2)C_(r)+.^(n-3)C_(r)+.........+.^(r)C_(r)=.^(n)C_(r+1)`.

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

Prove that : (""^(n)C_(r+1))/(""^(n)C_(r))=(n-r)/(r+1)

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

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

the roots of the equations |{:(.^(x)C_(r),,.^(n-1)C_(r),,.^(n-1)C_(r-1)),(.^(x+1)C_(r),,.^(n)C_(r),,.^(n)C_(r-1)),(.^(x+2)C_(r),,.^(n+1)C_(r),,.^(n+1)C_(r-1)):}|=0

Find the value of the determinant |(.^(n)C_(r-1),.^(n)C_(r),(r+1)^(n+2)C_(r+1)),(.^(n)C_(r),.^(n)C_(r+1),(r+2)^(n+2)C_(r+2)),(.^(n)C_(r+1),.^(n)C_(r+2),(r+3)^(n+2)C_(r+3))|

If .^(n)P_(r)=.^(n)P_(r+1) and .^(n)C_(r)=.^(n)C_(r-1) , find n and r.