There are 10 identical blocks of cuboid of dimension 2 inches × 3 inches × 5 inches. The blocks are kept one on top of the other in a random fashion to form a structure. How many structures with unique heights can be created using these blocks? (Neglect instability of the structure) NOTE: Two of the flat faces of each cube are parallel to the ground.

1) 29
2) 30
3) 31
4) 49
5) 32

• You can easily check by counting all the possible values: lowest possible value is (2)(10) = 20 and maximum possible value is (5)(10) = 50. So from 20 to 50 at maximum 31 different values are possible. that means you are left with only first two options. Now you need to check how many values between the above range are not possible and it is only one value i.e. 49 which is not possible to attain. Hence the answer is 2) 30.
  
  Now solving mathematically, required answer is just number of terms in the expansion of (x2 + x3 + x5)10
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