Combination: 3 boys with 7 chairs

[Michael_Light]

Homework Statement

Suppose there is 7 chairs arranged in a straight line, each of the 3 boys will sit randomly on one of the chair . In how many ways the boys can be seated if the 3 boys cannot sit next to each other? Assume that the boys are indistinguishable.

I listed out all the possible outcomes (which is 10), but i believe there is a generalized way to find the answer. Can anyone enlighten me?

Homework Equations

The Attempt at a Solution

Let O represent seat occupied by the boys and X is empty seat.

Possible outcomes:

XOXOXOX
XOXOXXO
XOXXOXO
XXOXOXO
OXXOXOX
OXOXXOX
OXOXOXX
OXXOXXO
OXOXXXO
OXXXOXO

 

[Ray Vickson]

Homework Statement

Suppose there is 7 chairs arranged in a straight line, each of the 3 boys will sit randomly on one of the chair . In how many ways the boys can be seated if the 3 boys cannot sit next to each other? Assume that the boys are indistinguishable.

I listed out all the possible outcomes (which is 10), but i believe there is a generalized way to find the answer. Can anyone enlighten me?

Homework Equations

The Attempt at a Solution

Let O represent seat occupied by the boys and X is empty seat.

Possible outcomes:

XOXOXOX
XOXOXXO
XOXXOXO
XXOXOXO
OXXOXOX
OXOXXOX
OXOXOXX
OXXOXXO
OXOXXXO
OXXXOXO

Using 'b' for 'boy' and 'e' for 'empty', start with bebeb and just figure out how many ways to add the two remaining 'e's.

 

[haruspex]

For a generalized approach, suppose C chairs and B boys, same restriction. Each occupied chair, except the rightmost, must have a vacant chair on its right. To handle that exception, introduce an extra chair on the right, guaranteed vacant. So we can pair up each occupied chair with that adjacent vacant chair, making B such pairs and C+1-2B other vacant chairs. Can you proceed from there?

 

[Michael_Light]

For a generalized approach, suppose C chairs and B boys, same restriction. Each occupied chair, except the rightmost, must have a vacant chair on its right. To handle that exception, introduce an extra chair on the right, guaranteed vacant. So we can pair up each occupied chair with that adjacent vacant chair, making B such pairs and C+1-2B other vacant chairs. Can you proceed from there?

Got it. Your hint is very useful. Thanks.