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# let x,y,z be three non-negative integer such that x+y+z=10 the maximum possible value of xyz+xy+yz+zx= ?

Nishant Vora IIT Patna
6 years ago
xyz+xy+xz+yz can be expressed as (x + 1)(y + 1)(z + 1) - (x + y + z + 1)

Since x+y+z=10

xyz+xy+xz+yz = (x + 1)(y + 1)(z + 1) - 11

So now we have to maximize
(x + 1)(y + 1)(z + 1) - 11

Hello Student, Please find the solution

Now x+y+z=10 can be written as (x+1) + (y+1)+ (z+1)=13

We can now use AM-GM Rule to get max value, which we will get when value of x+1,y+1 & z+1 is as close as possible (if not equal)

Now floor(13/3) = 4 so we can have max value at (4,4,(13-(4+4))
i.e. 4,4,5

So Max value = 4*4*5 - 11 = 69