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Step-by-Step Solution
Step 1: Understand the Problem
We have the set $S = \{1, 2, 3, \ldots, 20\}$. A subset $B \subseteq S$ is called “nice” if the sum of its elements is $203$. We want to find the probability that a randomly chosen subset of $S$ is “nice”.
Step 2: Total Number of Subsets
The set $S$ has $20$ elements. Therefore, the total number of subsets of $S$ (including the empty set and $S$ itself) is $2^{20}$.
Step 3: Sum of All Elements in S
The sum of the elements in $S$ is:
$$1 + 2 + 3 + \cdots + 20 = \frac{20 \times 21}{2} = 210.$$
For a subset $B$ to have sum $203$, the elements not in $B$ should sum to $210 - 203 = 7.$
Step 4: Find All Subsets Whose Elements Sum to 7
We now look for distinct elements in $S$ that add up to $7$. These elements correspond to the complement of the “nice” subset $B$. The possible combinations (using distinct elements from 1 to 20) that sum to $7$ are:
$\{7\}$
$\{1, 6\}$
$\{2, 5\}$
$\{3, 4\}$
$\{1, 2, 4\}$
Thus, there are $5$ distinct ways to form a subset of $S$ whose sum is $7$. Correspondingly, there are $5$ “nice” subsets whose sum is $203$.
Step 5: Calculate the Probability
Since there are $2^{20}$ total subsets, and $5$ of those subsets are “nice,” the probability of choosing a “nice” subset randomly is:
$$\frac{5}{2^{20}}.$$
Step 6: Final Answer
The probability that a randomly chosen subset of $S$ sums to $203$ is
$$\frac{5}{2^{20}}.$$
