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# There are 5 boys and 4 girls in my class. All of them are distinguishable.

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There are 5 boys and 4 girls in my class. All of them are distinguishable.

In how many ways can they be seated in a row of 9 chairs such that at least 2 boys are next to each other?

Guest Mar 5, 2015

#1
+27057
+5

First consider the number of ways they can be seated so that no boys are next to each other.  The only way this can happen is if the boys are on seats 1, 3, 5, 7 and 9.  There are 5 boys who could go on seat 1.  For each of these there are 4 boys who could go on seat 3 ...etc., so there are 5! ways of arranging the 5 boys on these seats.  For each of these there are 4! ways of arranging the girls on seats 2, 4, 6 and 8.  So in total there are 5!*4! ways of arranging everyone on 9 chairs such that no boys sit together.

Now there are 9! ways of arranging everyone on the 9 chairs, regardless of who sits next to whom.

This means there must be 9! - 5!*4! ways of arranging everyone on the 9 chairs such that at least two boys are next to each other.

$${\mathtt{9}}{!}{\mathtt{\,-\,}}{\mathtt{5}}{!}{\mathtt{\,\times\,}}{\mathtt{4}}{!} = {\mathtt{360\,000}}$$

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Alan  Mar 5, 2015
#1
+27057
+5

First consider the number of ways they can be seated so that no boys are next to each other.  The only way this can happen is if the boys are on seats 1, 3, 5, 7 and 9.  There are 5 boys who could go on seat 1.  For each of these there are 4 boys who could go on seat 3 ...etc., so there are 5! ways of arranging the 5 boys on these seats.  For each of these there are 4! ways of arranging the girls on seats 2, 4, 6 and 8.  So in total there are 5!*4! ways of arranging everyone on 9 chairs such that no boys sit together.

Now there are 9! ways of arranging everyone on the 9 chairs, regardless of who sits next to whom.

This means there must be 9! - 5!*4! ways of arranging everyone on the 9 chairs such that at least two boys are next to each other.

$${\mathtt{9}}{!}{\mathtt{\,-\,}}{\mathtt{5}}{!}{\mathtt{\,\times\,}}{\mathtt{4}}{!} = {\mathtt{360\,000}}$$

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Alan  Mar 5, 2015
#2
+93691
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I did it exactly the same way - that is comforting for me anyway

Melody  Mar 5, 2015