What is the sum of all binomial coefficients $(0 8) + (1 8) + (2 8) + \dots + (8 8)$ ?

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If $α$ and $β$ are the roots of the quadratic equation $x^{2} - 5 x + 6 = 0$ , what is the value of $α^{3} + β^{3}$ ?
What is the total number of terms in the algebraic expansion of $(x + y + z)^{10}$ ?
If the quadratic equation $x^{2} - (k - 2) x + (k + 1) = 0$ has equal real roots, what are the possible values of the constant $k$ ?
What is the coefficient of $x^{5}$ in the expansion of the product $(1 + x^{2})^{5} (1 + x)^{4}$ ?
If $α$ and $β$ are the roots of the quadratic equation $x^{2} - 2 x + 4 = 0$ , what is the value of $α^{6} + β^{6}$ ?
What is the term independent of $x$ in the binomial expansion of $(2 x + \frac{1}{3 x ^{2}})^{9}$ ?