Question

lications of probability and
dayshawn can choose two of his four t - shirts to take on a weekend trip.
if the t - shirts are labeled a, b, c, and d, which choice represents the sample space, s, for the event?
$s = \\{ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc\\}$
$s = \\{abcd, dcba\\}$
$s = \\{abcd\\}$
$s = \\{ab, ac, ad, bc, bd, cd\\}$

Explanation:

Step1: Understand the Problem

We need to find the sample space \( S \) when Dayshawn chooses 2 out of 4 t - shirts (labeled A, B, C, D). In combinatorics, when we choose a subset of items and the order of selection does not matter (since choosing shirt A then B is the same as choosing shirt B then A for a set of two shirts to take on a trip), we are dealing with combinations, not permutations.

Step2: Analyze Each Option

• Option 1: \( S=\{AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC\} \): This set considers the order of selection (permutations), but since the order of choosing two t - shirts for the trip does not matter (AB and BA represent the same set of two t - shirts), this is incorrect.
• Option 2: \( S = \{ABCD, DCBA\} \): This set represents choosing all 4 t - shirts, but the problem states that we are choosing 2 t - shirts, so this is incorrect.
• Option 3: \( S=\{ABCD\} \): This set represents choosing all 4 t - shirts, not 2, so this is incorrect.
• Option 4: \( S=\{AB, AC, AD, BC, BD, CD\} \): This set lists all the possible ways to choose 2 t - shirts from 4 where the order does not matter. For example, AB means choosing t - shirts A and B, AC means choosing A and C, etc. This is the correct representation of the sample space for choosing 2 out of 4 t - shirts when order does not matter.

Answer:

\( S = \{AB, AC, AD, BC, BD, CD\} \) (the last option in the given list of options)