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6.Permutation and Combination
hard
The number of triplets $(x, y, z)$. where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is
A
$80$
B
$114$
C
$92$
D
$136$
(JEE MAIN-2023)
Solution
$x+y+z=15$
Total no. of solution $={ }^{15+3-1} C _{3-1}=136$
Let $x = y \neq z$
$2 x + z =15 \Rightarrow z =15-2 t$
$\Rightarrow r \in\{0,1,2, \ldots 7\}-\{5\}$
$\therefore 7 \text { solutions }$
$\therefore 7$ solutions
$\therefore$ there are 21 solutions in which exactly
Two of $x _1 y _1 z$ are equal
There is one solution in which $x=y=z$
Required answer $=136-21-1=114$
Standard 11
Mathematics
