Question

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

(1) 80 

(2) 114 

(3) 92 

(4) 136

Answer 1

Correct option is (2) 114

x + y + z = 15

Total no. of solution = ^15+3-1C_3-1 = 136 ...(1)

Let x = y ≠ z

2x + z = 15 ⇒ z = 15 - 2t

⇒ r ∈ {0,1, 2,...7} - {5} 

∴ 7 solutions 

∴ there are 21 solutions in which exactly

Two of x₁ y₁ z are equal …(2) 

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

Required answer = 136 - 21 - 1 = 114