If $$\alpha $$ and $$\beta $$ be the coefficients of x⁴ and x² respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then

The coefficient of x⁷ in the expression
(1 + x)¹⁰ + x(1 + x)⁹ + x²(1 + x)⁸ + ......+ x¹⁰ is:

The greatest positive integer k, for which 49^k + 1 is a factor of the sum
49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

If ²⁰C₁ + (2²) ²⁰C₂ + (3²) ²⁰C₃ + ..... + (20² ) ²⁰C₂₀ = A(2^{$$\beta $$}), then the ordered pair (A, $$\beta $$) is equal to :