Question 170
How many quadruples of positive integers $(a, b, c, d)$ satisfy the inequalities
$$
a \gt b \gt c \gt d \quad \text{and} \quad \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \gt 2?
$$
$$
a \gt b \gt c \gt d \quad \text{and} \quad \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \gt 2?
$$