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Even numbers = $\{  2,4,6,8\}$

Odd numbers = $\{  1,3,5,7,9 \}$

Sum property =

Choosing $2$ numbers possibility = $\{EE,\color{Red}{EO},\color{Red}{OE},OO\}$

Example  $3+2=5$ Fail

Choosing $3$ number possibility = Use pigenhole no matter how you choose $3^{rd}$ number either will be $odd$ or $even$.
$\therefore$ Sum will be even.
Example $ {\{3,2\}}$ Choice $Odd$ or $Even $ lets say $5$ or $4$
Then , $5+3=8$ or $4+2=6$
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