Question

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 = {abcd} s = {abcd, dcba} s = {ab, ac, ad, bc, bd, cd} s = {ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc}

Explanation:

Step1: Understand the problem

We need to find the sample space for choosing 2 out of 4 t - shirts (A, B, C, D). The order of choosing does not matter here (choosing A then B is the same as choosing B then A).

Step2: Analyze each option

• Option 1: \(S = \{ABCD\}\) represents choosing all 4 t - shirts, not 2. So this is incorrect.
• Option 2: \(S=\{ABCD, DCBA\}\) also represents choosing all 4 t - shirts (in different orders), not 2. So this is incorrect.
• Option 3: \(S = \{AB, AC, AD, BC, BD, CD\}\). We list all combinations of 2 t - shirts. The number of combinations of \(n = 4\) items taken \(r=2\) at a time is given by the formula \(C(n,r)=\frac{n!}{r!(n - r)!}\). For \(n = 4\) and \(r=2\), \(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=\frac{4\times3}{2\times1}=6\). And the set \(\{AB, AC, AD, BC, BD, CD\}\) has 6 elements which are all the combinations of 2 t - shirts (order does not matter).
• Option 4: \(S=\{AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC\}\) considers order (since AB and BA are both included), but when choosing 2 t - shirts to take, the order of selection does not matter. So this is incorrect.

Answer:

\(S = \{AB, AC, AD, BC, BD, CD\}\) (the third option)