A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is _____.

$$a + b = c + d = S \text{ with distinct digits from } \{0,1,2,3,4,5,6,7\} \text{ containing } 7$$

$$(7, x) \text{ and } (y, z) \text{ where } 7+x = y+z$$

$$x=0, S=7 \implies (y,z) \in \{(1,6), (2,5), (3,4)\} \quad \text{(3 sets)}$$

$$x=1, S=8 \implies (y,z) \in \{(2,6), (3,5)\} \quad \text{(2 sets)}$$

$$x=2, S=9 \implies (y,z) \in \{(3,6), (4,5)\} \quad \text{(2 sets)}$$

$$x=3, S=10 \implies (y,z) \in \{(4,6)\} \quad \text{(1 set)}$$

$$x=4, S=11 \implies (y,z) \in \{(5,6)\} \quad \text{(1 set)}$$

$$\text{Total valid sets of 4 digits} = 3 + 2 + 2 + 1 + 1 = 9$$

$$\text{Arrangements per set} = 2 \text{ (pair swap)} \times 2! \text{ (first pair)} \times 2! \text{ (second pair)} = 8$$

$$\text{Maximum trials} = 9 \times 8 = 72$$