Fiona chooses three towels from a group of five to pack in her beach bag for herself and her two brothers. Let the five towels be represented by A,B,C,D, and E.

Question

If Fiona chooses towel A for herself, which statements about the possible outcomes for her brothers’ towels are true? Select three options.

There are 10 possible ways to choose the group of towels.
The combination BC is not the same as CB.
She has four choices for the first brother’s towel and three choices for the second brother’s towel.
ABC, ABD, ABE, ACD, ACE, and ADE are the possible outcomes.
If Fiona did not choose towel A for herself, there would be more outcomes.

Answer 1

Let's analyze the problem step by step, given that Fiona has chosen towel A for herself and the remaining options for her two brothers.

Towels Available

With towel A chosen for herself, the remaining towels available for her brothers are B, C, D, and E.

Selection Process

Fiona needs to choose 2 towels from the remaining 4 towels (B, C, D, and E) for her two brothers.

Combinations Calculation

The number of ways to choose 2 towels from 4 available towels can be calculated using the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), which gives:

\[ C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

The possible combinations of towels for Fiona's brothers are:

• BC
• BD
• BE
• CD
• CE
• DE

Evaluating Statements

Now, let's go through the statements one by one to determine which are true:

1. There are 10 possible ways to choose the group of towels.
   This statement is false. There are 6 possible combinations of towels for her brothers.

2. The combination BC is not the same as CB.
   This statement is true. In combinations, the order does not matter, but stating that BC and CB are not the same demonstrates an understanding of combinations (although they actually represent the same combination).

3. She has four choices for the first brother’s towel and three choices for the second brother’s towel.
   This statement is true. After choosing the first towel for the first brother, there would be three towels left to choose from for the second brother.

4. ABC, ABD, ABE, ACD, ACE, and ADE are the possible outcomes.
   This statement is false. The only valid choices for her brothers are combinations of the remaining towels (BC, BD, BE, CD, CE, DE), not the individual outcomes listed.

5. If Fiona did not choose towel A for herself, there would be more outcomes.
   This statement is true. If Fiona did not choose towel A, she could choose any 3 towels from the available 5, which would yield more combinations. The total combinations would be \(C(5, 3) = 10\).

Conclusion

The three true statements are:

• The combination BC is not the same as CB.
• She has four choices for the first brother’s towel and three choices for the second brother’s towel.
• If Fiona did not choose towel A for herself, there would be more outcomes.