Final answer:
The group of 4 representatives can be formed in 28 ways if it must contain both the President and the Secretary, and in 112 ways if it must contain either the President or the Secretary but not both. Calculations are based on combination formulas.
Step-by-step explanation:
The number of different ways the group of 4 representatives can be formed from a club with a President, a Secretary, and 8 other members under the given conditions is calculated using combinatorics.
(a) Both the President and the Secretary
With both the President and the Secretary in the group, we need to choose the remaining 2 members from the 8 other members. This is a combination problem since the order of selection does not matter. The formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number to choose from, and k is the number of selections made.
For this case, C(8, 2) = 8! / (2! * (8 - 2)!) = 28 ways.
(b) Either President or the Secretary
For this scenario, we can form two separate groups: one with the President and one with the Secretary.
Group with the President: Choose 3 out of the remaining 9 members (8 others + Secretary): C(9, 3) = 84 ways.
Group with the Secretary: Choose 3 out of the remaining 9 members (8 others + President): Also 84 ways.
However, we should not double-count the scenario where neither the President nor the Secretary is part of the group. Since we counted this scenario twice (once in each group), we must subtract it once. The number of ways to form a group without the President and the Secretary is C(8, 3) = 56 ways.
The total number of ways hence is 84 + 84 - 56 = 112 ways.