Show that:`.^nC_(n-r)+3.^nC_(n-r+1)+3.^nC_(n-r+2)+^nC_(n-r+3)=^(n+3)C_r`.

""^(n)C_(n-r)+3.""^(n)C_(n-r+1)+3.""^(n)C_(n-r+2)+""^(n)C_(n-r+3)=""^(x)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))

If ([""^(n)C_(r) + 4*""^(n)C_(r+1) + 6*""^(n)C_(r+2)+ 4*""^(n)C_(r+3) + ""^(n)C_(r+4)])/([""^(n)C_(r) + 3. ""^(n)C_(r+1)+ 3*""^(n)C_(r+2) + ""^(n)C_(r +3)])=(n + lambda)/(r+lambda) the value of lambda is

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 value of 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)]| is n^2+n-2 b. 0 c. ^n+3C_(r+3) d. ^n C_(r-1)+^n C_r+^n C_(r+1)

If .^nC_(n-4)=15 , find .^nC_6

The coefficient of x^r[0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+.... +(x+2)^(n-1) is a.^n C_r(3^r-2^n) b.^n C_r(3^(n-r)-2^(n-r)) c.^n C_r(3^r+2^(n-r)) d. none of these

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)):}| is

If (1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a_r is a. (.^nC_2)^2 b. .^n C_r .^n C_(r+1) c. .^(2n) C_r d. .^(2n) C_(r+1)

In the expansion of (1 + x)^(n) (1 + y)^(n) (1 + z)^(n) , the sum of the co-efficients of the terms of degree 'r' is (a) .^(n^3)C_r (b) .^nC_(r^3) (c) .^(3n)C_r (d) 3.^(2n)C_r