Question

question 11
there are 3 cards in a hat: one is a king, one is a queen, and one is an ace. two cards are to be selected at random without
replacement. using a tree diagram, obtain the sample space for the experiment. list the elements that make up the sample space.
a kkq, kka, qak, aqa
b kq, ka, qk, qa, ak, aq
c kqa, qka, akq
d kk, kq, ka, qk, qq, qa, ak, aq, aa
5 points

Explanation:

Step1: Analyze the problem

We need to find the sample space for selecting 2 cards without replacement from 3 cards (King - K, Queen - Q, Ace - A).

Step2: Use the formula for permutations

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 3\) and \(r=2\), so \(P(3,2)=\frac{3!}{(3 - 2)!}=\frac{3!}{1!}=3\times2 = 6\)
The possible pairs (using the first letter of each card):

• If the first card is K:
• Second card Q: KQ
• Second card A: KA
• If the first card is Q:
• Second card K: QK
• Second card A: QA
• If the first card is A:
• Second card K: AK
• Second card Q: AQ

Answer:

B. KQ, KA, QK, QA, AK, AQ