let x,y,z be three non negative integers such that x+y+z=10. the maximum possible value of xyz+xy+yz+zxis

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Solution

$Taking the 3 no' s as : x + 1, y + 1, z + 1 A . M \geq GM \frac{x + 1 + y + 1 + z + 1}{3} \geq {(x + 1) (y + 1) (z + 1)}^{\frac{1}{3}} \Rightarrow (\frac{13}{3})^{3} \geq (x + 1) (y + 1) (z + 1) \Rightarrow xyz + xy + yz + zx + x + y + z + 1 \leq (\frac{13}{3})^{3} \Rightarrow xyz + xy + yz + zx \leq (\frac{13}{3})^{3} - 11 Equality holds when x = y = z but x + y + z = 10 and x, y, z are integers . So, we get maximum value of the expression when any two of x, y, z are equal to 3 and third is equal to 4 . Taking x = y = 3 and y = 4 xyz + xy + yz + zx = 69$

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