I am quite rusty on combinatorial formulas, but I think the following practical question has a combinatorial answer. This is not homework.
A company gives it's technicians two work days per week to work at home. In a team of four, a manager and three technicians, what assignment of days working at home will gurantee that at least one technician or the manager is NOT at home on every working day (Mon-Fri)? I.E., at least one person from the team of four is in the office every working day.
There are only 10 unique combinations of two weekdays (5 things taken 2 at a time), so I think the question becomes which of the combinations of two days among four employees satisfies the constraint that the count of any one of the days assigned to work at home among the four employees is less than 4 (thus guaranteeing that for any weekday at least one employee is NOT at home)?
If I haven't formulated this question correctly, please enlighten me and help cure my ignorance (and rusty mathematics).
[Edit] Based on one of the answers I have seen so far, I need to clarify two things:
- All four team members always have two scheduled days at home each week
- I would like to know if there is a way to generate combinations of scheduled days at home for the whole team which satisfy the constraint of at least one team member not at home each weekday