Question

(a) the 2011 baseball team at ohio state university consisted of 12 freshmen, 12 sophomores, 8 juniors, and 9 seniors. the coaches want to choose two players from each class to represent the team at a booster club banquet. how many different ways can they make this choice? the coaches can choose the players to represent the team at the booster club banquet in different ways. (b) the coaches also need to choose two players overall to visit elementary schools in the community, with each player going to a different school. how many different ways can they make this choice? the coaches can choose the players to visit the elementary schools in different ways.

Explanation:

Step1: Calculate total number of players

Total number of players \(n=12 + 12+8 + 9=41\)

Step2: Use combination formula for part (a)

The number of ways to choose \(r = 2\) players out of \(n=41\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\). Here, \(C(41,2)=\frac{41!}{2!(41- 2)!}=\frac{41\times40\times39!}{2\times1\times39!}=\frac{41\times40}{2}=820\)

Step3: Use permutation formula for part (b)

The number of ways to assign \(2\) players to different schools (permutation as order matters). The formula for permutations \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 41\), \(r=2\), so \(P(41,2)=\frac{41!}{(41 - 2)!}=41\times40 = 1640\)

Answer:

(a) \(820\)
(b) \(1640\)