Question

A train schedule shows that 2 days per week one train departs, 3 days per week two trains depart, and 2 days per week three trains depart. How many different ways can the schedule be made?

Answer 1

210

Explanation:

We have three different types of train departure days:

• One train departs - required for 2 days per week
• Two trains depart - required for 3 days per week
• Three trains depart - required for 2 days per week

If we had a situation where all 7 days of the week had a distinguishable train schedule, we could order them in `7xx6xx5xx4xx3xx2xx1=7!` number of ways.

However, the schedules are grouped into 3 distinguishable groups but within the groups they are indistinguishable. We need to take out the numbers of ways that are created by the indistinguishable members of each group and we do that by dividing by the factorial of the count of each group. This gives us:

`(7!)/(2!3!2!)=(7xx6xx5xx4xx3!)/(3!xx2xx2)=7xx6xx5=210`