We need to find the following sums:

a+b a+c a+d b+c b+d c+d

Well logically if 7 is the smallest sum, then the 2 smallest numbers of our set MUST be the addends:

a+b=7 assume there is a c<a that implies c+b<7. Same argument applies to c<b.

So a+b = 7 leaves us with 1,6 2,5 and 3,4.

Additionally we know the 9 = a+c where c is the 3rd smallest number. If any other combinations combines to make 9, then a+(one of them) <9 and that number was not listed.

so if a=1,b=6, then c = 8. since 6+8=14, that leaves d=9 so a+d = 1+9= 10.

a+b=7

a+c=9

a+d=10

b+c=14

b+d=15

c+d=17

product = 1*6*8*9=432

if a=2 b =5 so c = 7 so a+c = 9. This however leaves b+c=12 which is smaller than 14 and not listed.

if a=3 b=4 c =6 since 4+6=10. a+d=14 d=11. If any other pair adds to 14. than a+d < 14 and would have shown up on our list.

a+b=7

a+c=9

a+d=14

b+c=10

b+d=15

c+d = 17

3*4*6*11 = 792 Our largest possible product