Question 1210229
Solution:
We are asked to find the number of subsets of $S = \{1, 3, 8, 17, 30, 36, 47, 58\}$ such that the sum of the elements in the subset is less than 20.

We can consider each element of $S$ and decide whether to include it in a subset. Since the elements are sorted, we can proceed by considering smaller elements first.

Case 1: Subsets with 0 elements
The only subset with 0 elements is the empty set, $\emptyset$. The sum of its elements is 0, which is less than 20.
Number of subsets = 1 ($\emptyset$)

Case 2: Subsets with 1 element
The subsets with 1 element are $\{1\}, \{3\}, \{8\}, \{17\}, \{30\}, \{36\}, \{47\}, \{58\}$.
The sums of the elements are 1, 3, 8, 17, 30, 36, 47, 58.
The sums less than 20 are 1, 3, 8, 17.
The subsets are $\{1\}, \{3\}, \{8\}, \{17\}$.
Number of subsets = 4

Case 3: Subsets with 2 elements
We consider pairs of distinct elements from $S$ and check if their sum is less than 20.
Pairs involving 1: $\{1, 3\} \rightarrow 4$, $\{1, 8\} \rightarrow 9$, $\{1, 17\} \rightarrow 18$. (3 subsets)
Pairs involving 3 (excluding pairs with 1): $\{3, 8\} \rightarrow 11$, $\{3, 17\} \rightarrow 20$ (sum not less than 20). (1 subset)
Pairs involving 8 (excluding pairs with 1 and 3): $\{8, 17\} \rightarrow 25$ (sum not less than 20). (0 subsets)

The 2-element subsets with a sum less than 20 are $\{1, 3\}, \{1, 8\}, \{1, 17\}, \{3, 8\}$.
Number of subsets = 4

Case 4: Subsets with 3 elements
We consider triplets of distinct elements from $S$ and check if their sum is less than 20.
Triplets involving 1 and 3: $\{1, 3, 8\} \rightarrow 12$, $\{1, 3, 17\} \rightarrow 21$ (sum not less than 20). (1 subset)
Triplets involving 1 and 8 (excluding triplets with 3): $\{1, 8, 17\} \rightarrow 26$ (sum not less than 20). (0 subsets)

The 3-element subset with a sum less than 20 is $\{1, 3, 8\}$.
Number of subsets = 1

Case 5: Subsets with 4 or more elements
The smallest 4 elements of $S$ are 1, 3, 8, 17. Their sum is $1 + 3 + 8 + 17 = 29$, which is not less than 20. Therefore, there are no subsets with 4 or more elements whose sum is less than 20.

Total number of subsets with a sum less than 20 is the sum of the number of subsets from each case:
Total = (Number of 0-element subsets) + (Number of 1-element subsets) + (Number of 2-element subsets) + (Number of 3-element subsets)
Total = $1 + 4 + 4 + 1 = 10$

Final Answer: The final answer is $\boxed{10}$