Integer solutions of the following question: How many integer solutions are there in this equation:
$$x_1 + x_2 + x_3 + x_4 + x_5 = 63, \quad x_i \ge 0, \quad x_2 ≥10$$
I got $C(56,3)$.   
Is that correct?

Substitute $y_1 = x_1 + 1, y_2 = x_2 - 9, y_3 = x_3 + 1, y_4 = x_4 + 1, y_5 = x_5 + 1$.  This yields the equivalent problem
$$
(y_1 - 1) + (y_2 + 9) + (y_3 - 1) + (y_4 - 1) + (y_5 - 1) = 63; y_i \ge 1
$$
i.e.
$$
y_1 + y_2 + y_3 + \cdots + y_5 = 58
$$
By a classic stars-and-bars argument (writing $58$ in unary) the answer is then
$$
{58 - 1 \choose 5 - 1} = {57 \choose 4}.
$$
