The number of triplets (x, y, z), where x, y, z are distinct non negative integers satisfying x + y + z = 15, is [2023]

Question

The number of triplets (x, y, z), where x, y, z are distinct non negative integers satisfying x + y + z = 15, is [2023]

Smart Achievers · Accepted Answer

(2)

$x + y + z = 15$

$Total number of solutions =^{15 + 3 - 1} C_{3 - 1} = 136 ...(1)$

$Let x = y \neq z; 2 x + z = 15 \Rightarrow z = 15 - 2 x$

$\Rightarrow z \in {0, 1, 2, \dots, 7} - {5}$ $∴ There are 7 solutions.$

$∴ There are 21 solutions in which exactly two of x, y, z are equal. ...(2)$

$There is one solution in which x = y = z . ...(3)$

$Required answer = 136 - 21 - 1 = 114$

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