The roots of the equation  xCr n−1Cr n−1Cr−1 x+1Cr nCr nCr−1 x+2Cr n+1Cr n+1Cr−1=0 are

xCr n−1Cr nCr x+1Cr nCr n+1Crx+2Cr n+1Cr n+2Cr=0           (1)⇒x!r!(x−r)!(n−1)!r!(n−r−1)!n!r!(n−r)!(x+1)!r!(x+1−r)!n!r!(n−r)!(n+1)!r!(n−r+1)!(x+2)!r!(x+2−r)!(n+1)!r!(n+1−r)!(n+2)!r!(n−r+2)!=0Taking x!r!(x−r)! common from C1, we have quadratic equation in x.Now in (1), if we put x = n - 1, C1 and C2 are the same; hence, x = n - 1 is one root of the equation.If we put x = n, then C1 and C3 are same. Hence, x = n is the other root.