The number of solutions of the equation x+y+z | vedclass

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Question

(A) Given the equation $x+y+z=10$, where $x, y, z \in \mathbb{Z}^+$.This is a problem of finding the number of positive integer solutions to the equat

Vedclass · Accepted Answer

$36$

Vedclass · Answer

(A) Given the equation $x+y+z=10$, where $x, y, z \in \mathbb{Z}^+$.This is a problem of finding the number of positive integer solutions to the equation $x_1+x_2+\dots+x_r=n$.The formula for the number of positive integer solutions is given by $^{n-1}C_{r-1}$.Here, $n=10$ and $r=3$.Therefore, the number of solutions $= ^{10-1}C_{3-1} = ^{9}C_{2}$.$^{9}C_{2} = \frac{9 	imes 8}{2 	imes 1} = 36$.

The number of solutions of the equation $x+y+z=10$, where $x, y$, and $z$ are positive integers, is:

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