Question

the local football team selects three captains to represent its team during the coin toss: one from the offense, one from the defense, and one from the special teams.

if the coaches decide to choose a quarterback and either the kicker or the punter to be two of the three captains, how many different groups of captains can the coaches select with these constraints?

offense		defense		special teams
running backs	3	linebackers	9	punters	1
receivers	6	cornerbacks	5	long snapper	1
linemen	9	safeties	3	returners	3
total	21	total	27	total	6

a 1,134

b 162

c 27

d 243

Explanation:

Step1: Determine number of quarterbacks

From the table, number of quarterbacks (offense) is 3.

Step2: Determine number of kickers and punters

Number of kickers (special teams) is 1, number of punters (special teams) is 1. So total for kicker or punter: \(1 + 1 = 2\).

Step3: Determine number of defense players

Total defense players is 27.

Step4: Calculate total groups

We need to choose 1 quarterback, 1 (kicker or punter), and 1 defense player. Using multiplication principle: \(3\times2\times27\)
First, \(3\times2 = 6\), then \(6\times27 = 162\). Wait, no, wait: Wait, kicker is 1, punter is 1. So cases: quarterback (3) kicker (1) defense (27) + quarterback (3) punter (1) defense (27). So \(3\times1\times27 + 3\times1\times27 = 81 + 81 = 162\)? Wait, no, wait: Wait, the problem says "a quarterback and either the kicker or the punter to be two of the three captains". So the three captains are: 1 from offense (quarterback), 1 from special teams (kicker or punter), and 1 from defense. So number of choices: quarterbacks (3) (kickers + punters) (1 + 1 = 2) defense (27). So \(3\times2\times27 = 162\). Wait, but let's check again. Wait, kicker is 1, punter is 1. So for kicker: 3 (QBs) 1 (kicker) 27 (defense) = 81. For punter: 3 (QBs) 1 (punter) 27 (defense) = 81. Total: 81 + 81 = 162. So the answer should be 162, which is option B.

Wait, but wait, maybe I made a mistake. Wait, the defense total is 27? Let's check the table: defense total is 27. Yes. Quarterbacks: 3. Kicker:1, Punter:1. So the number of ways: choose QB (3), choose special teams (kicker or punter: 2 options), choose defense (27). So 3227 = 162. Yes. So the correct answer is B.

Answer:

B. 162