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Solution


Q.

The number of triplets (x, y, z), where x, y, z are distinct non negative integers satisfying x + y + z = 15, is             [2023]
1 80  
2 114   
3 92  
4 136  

Ans.

(2)

x+y+z=15 

Total number of solutions= 15+3-1C3-1=136  ...(1) 

Let x=y≠z;   2x+z=15⇒z=15-2x 

⇒   z∈{0,1,2,…,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