Final answer:
There are 28 different ways to form a group of 4 representatives from a club that includes the President and the Secretary plus 8 other members, when the group must contain both the President and the Secretary.
Step-by-step explanation:
To calculate the number of different ways in which a group of 4 representatives can be formed from a club that contains the President, the Secretary and 8 other members, given that both the President and the Secretary must be included, we can use the principles of combinatorial math.
Since the President and the Secretary are already chosen, we need to pick 2 more members from the remaining 8 members. This is a combination problem because the order in which we select the members does not matter.
The number of ways to choose 2 members from 8 is given by the combination formula C(n, k) = n! / k!(n-k)! where n is the total number of items to choose from, and k is the number of items to pick.
In this case, n=8 and k=2, so the formula would be C(8, 2) = 8! / 2!(8-2)! = (8 * 7) / (2 * 1) = 28.
Therefore, the club can send representatives to the conference in 28 different ways if the group must include both the President and the Secretary.