Six people attend the theater together and sit in a row with

Six people attend the theater together and sit in a row with exactly 6 seats.

(a) If they shake hands with each other, how many handshakes will happen?

(b) How many ways can they be seated together in the row?

(c) Suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat?

(d) Suppose the six people consist of 3 married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?

Solution

a)

2 people in a handshake.

Thus, there are

6C2 = 15 handshakes. [answer]

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b)

There are

n! = 6! = 720 ways to seat them in a row.

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c)

We fix the doctor on one aisle, and permute the other 5.

There are 5! = 120 ways to do that.

As there are 2 aisles, there are 2*120 = 240 ways to do this. [answer, 240]

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d)

We can treat each couple as just one object.

Thus, there are 3 objects here, so there are

3! = 6 ways to permute them. [answer, 6]