Question:

Step 1: Determine the total number of ways to select three distinct numbers. The set of numbers is $\{-1, 0, 1, 2, 3\}$. We are selecting three numbers, $a, b, c$, without repetition. The order of selection does not matter for the set of numbers chosen. Thus, the total number of ways to select three distinct numbers from a set of five is given by the combination formula $\binom{n}{k}$: $$ \text{Total selections} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 $$ Step 2: Determine the number of favorable selections. We need to find the number of subsets $\{a, b, c\}$ such that their sum $a+b+c=0$. By examining all possible combinations of three distinct numbers from the set $\{-1, 0, 1, 2, 3\}$, it is found that there are 3 such subsets whose elements sum to zero. Step 3: Calculate the probability. The probability that $a+b+c=0$ is the ratio of the number of favorable selections to the total number of selections: $$ P(a+b+c=0) = \frac{\text{Number of favorable selections}}{\text{Total number of selections}} = \frac{3}{10} $$