Question

Let a be a factor of 120, then the number of integral solution of x1x2x3 = a is 

Solution

Correct option is

320

Let x4 be such that x4 = 120/a. then, the number of positive integral solutions of x1x2x3 = a is same as that of number of positive integral solutions of x1x2x3x4 = 120 = 23 × 3 × 5.     

We can assign 3 and 5 to unknown quantities in 4 × 4 ways. We can assign all 2 to one unknown in 4C1 ways, to two unknowns in (4C2)(2) and to three unknown in 4C3 ways. Hence, the number of required solutions  

          = 4 × 4 × [4C1 + (4C2)(2) + 4C3]   

          = 4 × 4 × 20 = 320.

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