There are 14,872 groups of 8 that can be made from a group of 16, considering the given conditions.
How to calculate the number of groups of 8 that can be made from a group of 16
To calculate the number of groups of 8 that can be made from a group of 16, considering the given conditions.
Case 1: Spencer is selected and Jensen is not selected.
In this case, we have 1 person (Spencer) already selected. We need to select 7 more people from the remaining 15 (excluding Spencer and Jensen).
The number of ways to select 7 people from a group of 15 is denoted as "15 choose 7" and can be calculated using the combination formula: C(15, 7) = 6435.
Case 2: James is selected and Jensen is not selected.
In this case, we have 1 person (James) already selected. We need to select 7 more people from the remaining 15 (excluding James and Jensen).
The number of ways to select 7 people from a group of 15 is again "15 choose 7" and equals 6435.
Case 3: Jensen is selected and both Spencer and Foley are not selected.
In this case, we have 1 person (Jensen) already selected. We need to select 6 more people from the remaining 15 (excluding Jensen, Spencer, and Foley).
The number of ways to select 6 people from a group of 15 is "15 choose 6" and can be calculated as C(15, 6) = 5005.
However, subtract the number of groups where both Jensen and Foley are selected since it violates the given condition.
To calculate this, find the number of groups where Jensen and Foley are both selected and subtract it from the total.
Case 4: Jensen and Foley are both selected.
In this case, we have 2 people (Jensen and Foley) already selected. We need to select 6 more people from the remaining 14 (excluding Jensen, Foley, Spencer, and James).
The number of ways to select 6 people from a group of 14 is "14 choose 6" and can be calculated as C(14, 6) = 3003.
Now, to calculate the total number of groups of 8 that can be made, sum up the results from the first three cases and subtract the result from Case 4:
6435 + 6435 + 5005 - 3003 = 14872
Therefore, there are 14,872 groups of 8 that can be made from a group of 16, considering the given conditions.