Introduction:
To choose a team of 8 players from a group of 12 players, we need to calculate the number of ways in which this selection can be made. Additionally, one player needs to be elected as the captain and another player as the vice captain.

Solution:
Let's break down the problem into two parts:

1. Selecting the team:
To choose a team of 8 players from a group of 12 players, we need to calculate the number of combinations. This can be done using the combination formula, which is given by:
nCr = n! / (r!(n-r)!)

In this case, n = 12 (total number of players) and r = 8 (number of players to be selected).

Using the combination formula, we can calculate the number of ways to select the team:
12C8 = 12! / (8!(12-8)!)
= (12 * 11 * 10 * 9 * 8!) / (8! * 4 * 3 * 2 * 1)
= (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1)
= 495

So, there are 495 ways to select a team of 8 players from a group of 12 players.

2. Electing the captain and vice captain:
Once the team is selected, one player needs to be elected as the captain and another player as the vice captain. Since any of the 8 selected players can be chosen for these roles, we have 8 options for the captain and 7 options for the vice captain.

Therefore, the number of ways to elect the captain and vice captain is:
8 * 7 = 56

Total number of ways:
To calculate the total number of ways to choose the team and elect the captain and vice captain, we multiply the number of ways for each step:
Total ways = Number of ways to select the team * Number of ways to elect the captain and vice captain
= 495 * 56
= 27720

Therefore, there are 27720 ways to choose a team of 8 players from a group of 12 players, with one player elected as the captain and another player as the vice captain.

12C8×8C1