Question

In a small town in upstate New York, there are 4 bakeries, which are each closed one day per week.

1. How many ways are there for the bakeries to choose the days when they are closed?
2. Same question as 1. for the case when two bakeries cannot be closed on the same day
3. Same question as 1. for the case when at least one bakery is open on any given day of the week

Answer 1

1) Number of ways for the bakeries to choose the days when they are closed is computed here as:

= 7*7*7*7

= 7⁴ as each bakery has 7 days out of which it would be closed on 1 of them

= 2401

Therefore there are 2401 ways to choose the days when the bakery would be closed.

2) Given that the 2 bakeries cannot be closed on the same day, the number of ways here is computed as the number of permutation of 7 days taken 4 at a time

= 7*6*5*4

= 840

Therefore there are 840 ways to choose the days when the bakery would be closed given that two of them cannot be closed on the same day.

3) Given that at least one bakery is open on any given day of the week, we get here:
= Total number of ways to choose the days when the bakery would be closed - Total number of ways that all bakeries are closed on the same day

= 7⁴ - 7

= 2394

Therefore there are 2394 ways to choose the days here.