What I’m interpreting is we have to find number of subsets of size 5 that contains atleast 2 even numbers.
Even numbers = $\{0, 2,4, 6, 8\}$
Odd numbers = $\{1, 3, 5, 7, 9\}$
Total #subsets of size $5$ which contains atleast $2$ even number = #subset contain $2$ even and $3$ odd + #subset contain $3$ even and $2$ odd + #subset contain $4$ even and $1$ odd + #subset contain $5$ even and $0$ odd
$= \large{5\choose 2} *\large{5\choose 3} + \large{5\choose 3} * \large{5\choose 2} + \large{5\choose 4} * \large{5\choose 1} + \large{5\choose 5} * \large{5\choose 0}$
$= 10*10 + 10*10 + 5*5 + 1*1 = 100+100+25+1 = 226$
Clearly above answer is not given, so assuming $5$ is misprinted.
We have to find number of subsets containing atleast 2 even number.
Odd numbers have 2 choices either they can be in set or not. For even numbers there can be either $2$ or $3$ or $4$ or $5$ even numbers in the subset.
$\therefore \text{#subsets} = 2^5 * {\large{( {5\choose 2} + {5\choose 3} + {5\choose 4} + {5 \choose 5})}} = 2^5 *(10 + 10 + 5 +1) = 32*26 = 832$
