`.^(n)c_(r)+^(n'-1)c_(r-1)+^(n-1)c_(r-2)=^(n+1)c_(r)`

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))

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

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

Prove that .^(n)C_(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)C_(r)),(.^(x+1)C_(r),,.^(n)C_(r),,.^(n+1)C_(r)),(.^(x+2)C_(r),,.^(n+1)C_(r),,.^(n+2)C_(r)):}|=0

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

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) + .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

The expression ""^(n)C_(r)+4.""^(n)C_(r-1)+6.""^(n)C_(r-2)+4.""^(n)C_(r-3)+""^(n)C_(r-4)

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

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r for which S = .^(20)C_(r.).^(10)C_(0)+.^(20)C_(r-1).^(10)C_(1)+"........".^(20)C_(0).^(10)C_(r) is maximum can not be