Question

how many different groups of captains can the coaches select? chart with offense (quarterbacks: 3, running backs: 3, receivers: 6, linemen: 9, total: 21), defense (linemen: 9, linebackers: 9, cornerbacks: 5, safeties: 3, total: 27), special teams (kickers: 1, punters: 1, long snapper: 1, returners: 3, total: 6) a 243 b 3,402 c 567 d 54

Explanation:

Step1: Identify the principle

This is a problem of counting the number of ways to select one captain from each of the three groups (Offense, Defense, Special Teams). We use the multiplication principle of counting, which states that if there are \(m\) ways to do one thing, \(n\) ways to do a second thing, and \(p\) ways to do a third thing, then there are \(m\times n\times p\) ways to do all three things together.

Step2: Get the number of choices for each group

From the table, the total number of players in Offense is \(21\), in Defense is \(27\), and in Special Teams is \(6\).

Step3: Apply the multiplication principle

To find the number of different groups of captains, we multiply the number of choices from each group. So we calculate \(21\times27\times6\).

First, calculate \(21\times27\):
\(21\times27 = (20 + 1)\times27=20\times27+1\times27 = 540+27 = 567\)

Then, multiply the result by \(6\):
\(567\times6=3402\)

Answer:

B. 3,402