# How do I find the number of ways people can be seated on a bus?

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30 people are to travel by bus. It can carry 17 passengers outside and 13 inside. 4 refuse to travel outside and 5 refuse to travel inside. How many ways can the passengers be seated?

30 people are to travel by bus. It can carry 17 passengers outside and 13 inside. 4 refuse to travel outside and 5 refuse to travel inside. How many ways can the passengers be seated?

##### 1 Answer

#### Explanation:

I'm going to assume that we're talking about an open-air double decker bus - one that has seating upstairs (with no roof) and downstairs (and so "inside").

We have a bus that will be full - we have 30 people who are going to travel in a bus that has 30 seats. If we had no other considerations, there would be

However, we do have a couple of things to deal with. 4 people refuse to be outside (and so will be inside where there are 13 seats) and 5 people refuse to be inside (and so will be outside where there are 17 seats).

Let's work with these people first.

The **4 people who refuse to be outside** will sit somewhere inside where there are 13 seats. This is a permutation problem, in that we do care in what seats they sit, and so we have:

We can do the same thing with the **5 people who refuse to sit inside** - they'll be outside somewhere in the 17 available seats:

And now we can work with the **remaining passengers**. There are 21 people who will sit anywhere on the bus, and so anywhere amongst the remaining 21 seats. That's

In total then, we have: