Find k if `.^(n)C_R + ^(n)C_(r-1) = "^(k)C_r`

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

If ""^(n)C_(r ): ""^(n)C_(r+1)=1:2 and ""^(n)C_(r+1): ""^(n)C_(r+2)=2:3 , find n and r.

If .^(n)C_(r )=84, .^(n)C_(r-1)=36 " and" .^(n)C_(r+1)=126 , then find the value of n.

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

the value of overset(n)underset(r=2)(Simga)(-2)^(r ) |{:( .^(n-2)C_(r-2),,.^(n-2)C_(r-1),,.^(n-2)C_(r)),(-3,,1 ,,1),(2,,-1,,0):}| (n gt 2)

The vaule of sum_(r=0)^(n-1) (""^(C_(r))/(""^(n)C_(r) + ""^(n)C_(r +1))) is equal to

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

The number of values of 'r' satisfying the equation ""^(39)C_(3r-1)- ""^(39)C_(r^(2) )= ""^(39)C_(r^(2)-1) - ""^(39)C_(3r) is

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

If a + b = k, when a, b gt o and S(k, n) = sum _(r=0)^(n) r^(2) (""^(n) C_(r) ) a^(r) cdot b^(n-r) , then