In how many ways can you sit 5 people in a row of 20 seats...

Lucia Grimes

Lucia Grimes



In how many ways can you sit 5 people in a row of 20 seats if no 2 can sit together?
I've seen the simpler problem of just sitting 2 people in non consecutive seats. In that case, I would subtract from the total number of ways to sit the 2 persons the number of ways of sitting them together.
In this harder version of the problem,I've though of the same thing, but now considering the case were 2, 3, 4 or 5 sit together. But that seems to count duplicate cases.

Answer & Explanation




2022-07-15Added 13 answers

Step 1
Seat 5 people (A,B,C,D and E) first and then add 4 seats (denoted by s) between them so none of them are seated adjacent to each other. Now you are left with 11 seats to put in 6 places - 4 places between them or at two ends (denoted by ).
A s B s C s D s E
Step 2
Now the problem is equivalent to finding the number of 6-tuples of non-negative integers whose sum is 11 for which you use stars and bars method. Lastly, there are 5! ways to arrange people in the their seats.
Ellen Chang

Ellen Chang


2022-07-16Added 5 answers

Sit the 5 people in 16 seats, and then add a seat between each pair.
This gives ( 16 5 ) as the answer.

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