Question

A cuboid has volume 120 and each side is integer what is the number of cuboid possible

Answer 1

Since volume is product of length(L), breadth(B) and height(H) let us prime factorize 120

i.e 120 =  2*60

           = 2*2*30

           = 2*2*2*15

           = 2*2*2*3*5  

Hence 120 = 2*2*2*3*5

So there are 5 objects and we need to form 3 groups. This can be understood as selecting two lines among four lines (here lines acts as boundaries between numbers , creating two boundaries ensures three groups )

So number of ways = 4C2 = 6 ways

Here we did not consider 1 as a factor.

So if one of the dimensions is 1 then we need to form 2 groups (1 boundary) out of 5 objects

⇒ Number of ways = 4C1 = 4 ways

If two dimensions are 1, then we are left with the other dimension as 120. This is another way

So total number of ways are 6 + 4 + 1 = 11 ways

Here we consider the triads (2,15,4) and (2,4,15) as same because on roating the cuboid one  triad converts into other triad.

If the cuboid is immovable, then number of ways are 11 *3 = 33 (here we dont have even number of 2's or 3's or 5's else we need to eliminate repetitions araising out of it.)