(a) The 2011 baseball team at Ohio State University consisted of 10 freshmen, 10 sophomores, 12 juniors, and 12 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. Part 2 of 2 (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.

Step 1 ：Given that the 2011 baseball team at Ohio State University consisted of 10 freshmen, 10 sophomores, 12 juniors, and 12 seniors.

Step 2 ：For part (a), the coaches want to choose two players from each class to represent the team at a booster club banquet. This is a combination problem because the order of selection does not matter. We can use the combination formula \(C(n, k) = \frac{n!}{(n-k)!k!}\) where n is the total number of items, and k is the number of items to choose. We need to calculate this for each class and then multiply the results together because we are doing all of these actions (choosing two players from each class) simultaneously.

Step 3 ：Calculating for each class, we get: \(C(10, 2) = 45\) for freshmen, \(C(10, 2) = 45\) for sophomores, \(C(12, 2) = 66\) for juniors, and \(C(12, 2) = 66\) for seniors.

Step 4 ：Multiplying these results together, we get: \(45 * 45 * 66 * 66 = 8820900\).

Step 5 ：Thus, for part (a), the coaches can choose the players to represent the team at the booster club banquet in \(\boxed{8820900}\) different ways.

Step 6 ：For part (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. This is also a combination problem, but this time n is the total number of players on the team (10 freshmen + 10 sophomores + 12 juniors + 12 seniors = 44 players), and k is 2 because we are choosing two players.

Step 7 ：Calculating for the entire team, we get: \(C(44, 2) = 946\).

Step 8 ：Thus, for part (b), the coaches can choose the players to visit the elementary schools in \(\boxed{946}\) different ways.