For the natural numbers m, n, if $${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$ and $${a_1} = {a_2} = 10$$, then the value of (m + n) is equal to :

The coefficient of x²⁵⁶ in the expansion of

(1 $$-$$ x)¹⁰¹ (x² + x + 1)¹⁰⁰ is :

Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + .... + a₄₀x⁴⁰. Then a₁ + a₃ + a₅ + ..... + a₃₇ is equal to

The value of $$\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)} $$ is equal to :