Question

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 choosing B then A. This statement is true.
2. For "Each shirt can be paired with any one of other the remaining shirts": If there are 5 shirts, when choosing a pair, each shirt (e.g., A) can be paired with B, C, D, E (4 remaining shirts), and this holds for each shirt. This statement is true.
3. For "There are twenty possible ways to choose the group of shirts": The number of combinations of choosing 2 shirts from 5 is given by the combination formula \( C(n,r)=\frac{n!}{r!(n - r)!} \), where \( n = 5 \) and \( r=2 \). So \( C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4\times3!}{2\times1\times3!}=10 \), not 20. This statement is false.
4. For "He has five choices for the first shirt and five choices for the second shirt": For the first shirt, there are 5 choices, but for the second shirt, since one shirt is already chosen, there are \( 5- 1=4 \) choices (not 5). This statement is false.
5. For "If he chooses shirt B, there are four possible outcomes for choosing the second shirt": If shirt B is chosen first, the remaining shirts are A, C, D, E (4 shirts), so there are 4 choices for the second shirt. This statement 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.