Dayshawn can choose two of his four t-shirts to take on a weekend trip. If the [tex]$t$-shirts[/tex] are labeled [tex]$A, B, C$[/tex], and [tex]$D$[/tex], which choice represents the sample space, [tex]$S$[/tex], for the event?
[tex]$S={A B C D}$[/tex]
[tex]$S={A B C D, D C B A}$[/tex]
[tex]$S={A B, A C, A D, B C, B D, C D}$[/tex]
[tex]$S={A B, B A, A C, C A, A D, D A, B C, C B, B D, D B, C D, D C}$[/tex]
Answer (1)
The problem involves finding all possible combinations of choosing 2 t-shirts out of 4.
Since the order of selection does not matter, we use combinations.
List all possible pairs: AB, AC, AD, BC, BD, CD.
The sample space is { A B , A C , A D , BC , B D , C D } .
Explanation
Understand the problem Dayshawn has four t-shirts labeled A, B, C, and D. He needs to choose two of them for a weekend trip. We need to find the sample space S, which represents all possible combinations of two t-shirts he can choose.
Recognize combinations Since the order in which Dayshawn chooses the t-shirts doesn't matter (choosing A then B is the same as choosing B then A), we are looking for combinations, not permutations. This means that AB is the same as BA, AC is the same as CA, and so on.
List all possible pairs Let's list all the possible pairs of t-shirts Dayshawn can choose:
A and B (AB)
A and C (AC)
A and D (AD)
B and C (BC)
B and D (BD)
C and D (CD)
Determine the sample space Therefore, the sample space S is {AB, AC, AD, BC, BD, CD}.
Select the correct option Comparing this with the given options, we find that the correct choice is: S = { A B , A C , A D , BC , B D , C D }
Examples
In a class of four students (Alice, Ben, Cathy, and David), you want to form study groups of two. The sample space helps you list all possible unique study groups: {AliceBen, AliceCathy, AliceDavid, BenCathy, BenDavid, CathyDavid}. This ensures every possible pairing is considered without duplicates, which is crucial for fair group assignments or analyzing group dynamics.
