Final answer:
To determine the number of ways to form a group of 4 representatives including the president and the secretary from a club of 10 members, we calculate the combinations of choosing 2 additional members from the 8 remaining, resulting in 28 ways.
Step-by-step explanation:
The student has asked how many different ways a group of 4 representatives can be formed from a club of 10 members (1 president, 1 secretary, and 8 other members), if the group must contain both the president and the secretary.
Since both the president and the secretary must be in the group, we have 2 positions filled and need to choose 2 more representatives from the remaining 8 members. This is a combinatorial problem, specifically a combination since the order in which we choose the members does not matter.
Thus, we calculate the number of combinations of the remaining 2 positions from the 8 members using the combination formula:
C(n, k) = n! / [k! * (n-k)!]
Where n is the total number of items, and k is the number of items to choose.
Apply this formula:
C(8, 2) = 8! / [2! * (8-2)!] = (8 * 7) / (2 * 1) = 28 ways
Therefore, there are 28 ways to form a group of 4 representatives that include both the president and the secretary.