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.

Question

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.)

W.
X.

Y.
Z.

A.
X
B.
Z
C.
Y
D.
W

Answer 1

To determine the possible unique outcomes that Sarah can create by choosing three cards randomly at one time from the selected cards, we begin by outlining how many cards of each type she has:

• Spades (S): 5
• Clubs (C): 2
• Hearts (H): 1

We will analyze the combinations of these cards in terms of how many cards can be drawn from each suit.

1. All 3 from Spades: \( SSS \)
2. 2 from Spades and 1 from Clubs: \( SS C \)
3. 2 from Spades and 1 from Hearts: \( SS H \)
4. 1 from Spades and 2 from Clubs: \( S CC \)
5. 1 from Spades, 1 from Clubs, and 1 from Hearts: \( S C H \)
6. 1 from Spades and 2 from Hearts: (impossible since only 1 Heart exists)
7. All 3 from Clubs: (impossible since only 2 Clubs exist)
8. 2 from Clubs and 1 from Hearts: (impossible)

Now, let's list the possible unique outcomes based on viable combinations:

• 3 Spades:

  • \( SSS \)

• 2 Spades and 1 Club:

  • \( SSC \) (representing any combination of 2 spades and 1 club, as their identity isn't strictly necessary)

• 2 Spades and 1 Heart:

  • \( SSH \)

• 1 Spade and 2 Clubs:

  • \( S CC \)

• 1 Spade, 1 Club, and 1 Heart:

  • \( S C H \)

Now, we can summarize them:

1. \( SSS \)
2. \( SSC \)
3. \( SSH \)
4. \( S CC \)
5. \( S C H \)

If we have lists identified by W, X, Y, and Z in the question there would likely be certain selections.

If we could see the outcomes from each letter (W, X, Y, Z), we would match our outcomes to each list to determine which option (A, B, C, D) accurately represents all possible unique outcomes Sarah can achieve by choosing three cards from her selected eight cards.

Given the absence of the details of lists W, X, Y, and Z, I can't definitively match outcomes to those lists or answer which option is correct among A, B, C, and D without additional context. However, the unique outcomes we derived based on combinations of suited cards are as follows:

1. SSS
2. SSC
3. SSH
4. S CC
5. S C H

If provided with the outcomes as listed under the options W, X, Y, and Z, I can tell you which option among A, B, C, or D would represent the complete set of combinations correctly. Please provide those lists for definitive identification.