Final answer:
To divide 20 executives into foursomes, we use combinations to select groups and then permute these groups since order matters. For including board members, we repeat this process ensuring that each foursome includes one board member.
Step-by-step explanation:
When dividing a company of 20 executives into foursomes (groups of 4 people), we must consider that the order of selection within each foursome does not matter, but the order of the foursomes themselves does matter. To calculate the number of ways to do this, we use the concept of combination for each foursome, and then we use permutation to determine the order of the foursomes since they are distinct.
To solve part (a), we calculate the number of ways to select 4 people from 20, then 4 from 16, 4 from 12, and so on, for the five foursomes. The number of ways to divide them without regard to the order of the foursomes is simply the product of these combinations:
- 20 choose 4 for the first foursome
- 16 choose 4 for the second foursome
- 12 choose 4 for the third foursome
- 8 choose 4 for the fourth foursome
- 4 choose 4 for the fifth foursome
However, since the order of the foursomes matters, we must multiply the result by 5!, the number of ways to arrange these five distinct groups.
For part (b), each foursome must include one of the five board members, so we can select the remaining three members for each foursome from the remaining 15 executives. The five board members are already distinct, so their permutation is inherent in the question.
- 15 choose 3 for the remaining slots in the first foursome
- 12 choose 3 for the remaining slots in the second foursome
- 9 choose 3 for the remaining slots in the third foursome
- 6 choose 3 for the remaining slots in the fourth foursome
- 3 choose 3 for the remaining slots in the fifth foursome
Taking these calculations, you get the total number of ways to create the required foursomes respecting the constraints for each part of the question.