Question

Sarah selects eight cards from a pack of well shuffled cards. Five out of those eight cards are spades, two are clubs, and one is hearts. Which list shows all the possible unique outcomes if Sarah chooses three cards randomly at one time? (Note: One outcome is shown per row in the tables.)

Answer 1

To find all possible unique outcomes when Sarah chooses three cards randomly from the selection of eight cards (5 spades, 2 clubs, and 1 heart), we can break it down by the composition of her drawn cards. The cards can be represented as follows:

- S (spades) = 5
- C (clubs) = 2
- H (hearts) = 1

Since Sarah has 5 spades (S1, S2, S3, S4, S5), 2 clubs (C1, C2), and 1 heart (H), we need to consider various combinations of 3 cards drawn from these categories and list them systematically.

### Possible Unique Combinations of Three Cards:

1. **All Spades (SSS)**:
- S1, S2, S3
- S1, S2, S4
- S1, S2, S5
- S1, S3, S4
- S1, S3, S5
- S1, S4, S5
- S2, S3, S4
- S2, S3, S5
- S2, S4, S5
- S3, S4, S5

2. **Two Spades, One Club (SSC)**:
- S1, S2, C1
- S1, S2, C2
- S1, S3, C1
- S1, S3, C2
- S1, S4, C1
- S1, S4, C2
- S1, S5, C1
- S1, S5, C2
- S2, S3, C1
- S2, S3, C2
- S2, S4, C1
- S2, S4, C2
- S2, S5, C1
- S2, S5, C2
- S3, S4, C1
- S3, S4, C2
- S3, S5, C1
- S3, S5, C2
- S4, S5, C1
- S4, S5, C2

3. **Two Spades, One Heart (SSH)**:
- S1, S2, H
- S1, S3, H
- S1, S4, H
- S1, S5, H
- S2, S3, H
- S2, S4, H
- S2, S5, H
- S3, S4, H
- S3, S5, H
- S4, S5, H

4. **One Spade, Two Clubs (SCC)**:
- S1, C1, C2
- S2, C1, C2
- S3, C1, C2
- S4, C1, C2
- S5, C1, C2

5. **One Spade, One Club, One Heart (SCH)**:
- S1, C1, H
- S1, C2, H
- S2, C1, H
- S2, C2, H
- S3, C1, H
- S3, C2, H
- S4, C1, H
- S4, C2, H
- S5, C1, H
- S5, C2, H

6. **One Club, Two Hearts (CCH)**:
- There’s only one heart, so this combination does not apply.

### Summary of Outcomes:
In total, the combinations are:

- 10 outcomes of (SSS)
- 20 outcomes of (SSC)
- 10 outcomes of (SSH)
- 5 outcomes of (SCC)
- 10 outcomes of (SCH)

### Final Listing:

Here is the layout of some of these combinations summarized:

| Outcome |
|---------|
| S1, S2, S3 |
| S1, S2, S4 |
| S1, S2, S5 |
| S1, S3, S4 |
| ... |
| S1, C1, H |
| S2, C1, H |
| S3, C2, H |
| ... |
| C1, C2, H | (if applicable) |

The total possible unique outcomes can be obtained by following this systematic breakdown. If you need a complete printout of every combination, let me know.