Question

finding possible outcomes
eli chooses two shirts from a group of five to pack for a weekend trip. let each shirt be represented by a, b, c, d,
and e. which statements about the situation are true? check all that apply.
the combination of ab and ba are the same.
each shirt can be paired with any one of other the remaining shirts.
there are twenty possible ways to choose the group of shirts.
he has five choices for the first shirt and five choices for the second shirt.
if he chooses shirt b, there are four possible outcomes for choosing the second shirt.

Explanation:

1. For "The combination of AB and BA are the same": In combinations, the order does not matter. So choosing shirt A then B is the same as B then A. This is true.
2. For "Each shirt can be paired with any one of other the remaining shirts": When choosing 2 shirts, each shirt (e.g., A) can be paired with B, C, D, E (4 remaining shirts), so this is true as each has 4 possible pairs.
3. For "There are twenty possible ways to choose the group of shirts": Using combination formula \( C(n,r)=\frac{n!}{r!(n - r)!} \), where \( n = 5 \), \( r=2 \), \( C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10 \), not 20. So this is false.
4. For "He has five choices for the first shirt and five choices for the second shirt": After choosing the first shirt (5 choices), there are 4 remaining for the second, not 5. So this is false.
5. For "If he chooses shirt B, there are four possible outcomes for choosing the second shirt": If B is chosen, remaining shirts are A, C, D, E (4), so this is true.

Answer:

• The combination of AB and BA are the same.
• Each shirt can be paired with any one of other the remaining shirts.
• If he chooses shirt B, there are four possible outcomes for choosing the second shirt.