The first 6 bolts consist of one of each type.
That leaves 4 to pick.
If all 4 are of the same type there are 6 choices.
If two types are chosen, there are 15 ways of choosing two types out of 6.
With 4 bolts of 2 different types we can have 3 of one and 1 of the other, and we can have 2 of each. If the types are labelled A and B, we have AAAB, BBBA and AABB. So we have 3*15=45 different ways of choosing two bolts out of 6 types.
If three types are chosen, there are 20 ways of choosing 3 types out of 6.
With 4 bolts of 3 different types, we can have 2 of one and 1 each of the other two. If we use type labels A, B and C, we have AABC, BBAC, CCAB. So we have 3*20=60 different ways of choosing three bolts out of 6 types.
If 4 different types are chosen out of 6 there are 15 ways of selecting the bolts.
The total is therefore: 6+45+60+15=126.