Question

patty, quinlan, and rashad want to be club officers. the teacher who directs the club will place their names in a hat and choose two without looking. the student whose name is chosen first will be president and the student whose name is chosen second will be vice president. which choice represents the sample space, s, for this event? \\( \bigcirc \\ s = \\{pqr\\} \\) \\( \bigcirc \\ s = \\{pqr, prq, qpr, qrp, rpq, rqp\\} \\) \\( \bigcirc \\ s = \\{pq, pr, qr\\} \\) \\( \bigcirc \\ s = \\{pq, qp, pr, rp, qr, rq\\} \\)

Explanation:

Step1: Understand the Problem

We need to find the sample space for choosing two officers (president and vice - president) from three students: Patty (P), Quinlan (Q), and Rashad (R). The order matters here because the first - chosen is president and the second - chosen is vice - president.

Step2: List All Possible Outcomes

• If we start with P:
• If P is first (president), the second (vice - president) can be Q or R. So we have PQ and PR.
• If we start with Q:
• If Q is first, the second can be P or R. So we have QP and QR.
• If we start with R:
• If R is first, the second can be P or Q. So we have RP and RQ.

Combining all these possible ordered pairs, the sample space \(S=\{PQ, QP, PR, RP, QR, RQ\}\).

Step3: Eliminate Incorrect Options

• Option 1: \(S = \{PQR\}\) is incorrect because we are choosing only 2 names, not 3.
• Option 2: \(S=\{PQR, PRQ, QPR, QRP, RPQ, RQP\}\) is incorrect because we are choosing 2 names, not 3.
• Option 3: \(S = \{PQ, PR, QR\}\) is incorrect because it does not account for the reverse order (e.g., QP, RP, RQ) which are different outcomes since the order of selection (president and vice - president) matters.

Answer:

\(S = \{PQ, QP, PR, RP, QR, RQ\}\) (the fourth option)