Urgent?
there are the following ways three girls can be seated.
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So there are 7 ways at least three girls can be assorted. Because the girls are individuals, the number of ways to sort these seats are for girls is
7*4*3*2.
Now the ways to sort the remaining seats for one girl and five boys is..
6!
So my thinking is that you have for an answer the multiplication of these, or
7*4!*6!
Because
There are 5 boys and 4 girls in my class.
In how many ways can they be seated in a row of 9 chairs such that at least 3 girls are all next to each other?
4 answers
It's the number of ways you can put precisely 3 girls next to each other plus the number of ways you can put preciesly 4 next to each other.
To get precisely 3 girls next to each ther, you can choose the 3 girls and the order in which they will take their seats in 4! = 24 ways. There are 4! ways you can rearrange 4 girls, if we pick the first 3, you need to divide by the number of ways the girls we are not selcted can be rearranged, which is 1 in this case. Or you can say that you can choose the 3 girls in 4 ways, because this amounts to choosing the one who is going to be left out and then the 3 you choose can be rearranged in 3!= 6 ways and 3!*4 = 4!.
There are 7 seats available for each of these 24 possibilities. If we enumerate the seats from 1 to 9 counting from left to right, then the leftmost girl can take seat nrs 1 till 7. If she chooses seat nr. 1 or 7, then after the three girls are seated, out of the 6 remaining free seats, one seat will be excluded for the firth girl as she can't sit next to the other girls. There will then be 5 possible seats for her. So, in this case we have for each of the two possible seat choices for the leftmost girl, 4!*5 = 5! to get all the girls seated, so 2* 4!*5 = 2* 5! possibilities in total.
If the leftmost girl chooses seat nr. 2,3,4,5,or 6, then there are only 4 possible seats for the fourth girl. so 4!*4 ways to get the girls seated for each choice the leftmost girl makes, therefore 5*4*4! = 4*5! possibilities in total for these seat choices.
The total number of ways to get precisely 3 girls seated next to each other is thus 4*5! + 2*5! = 6*5! = 6!
Then there are 5! ways to get the five boys seated in the remaining seats, so there are 5!*6! ways to get the boys and girls seated such that there are precisely 3 girls sitting next to each other.
Then we need to consider the number of ways one can seat all the four grls next to each other. There are 4! ways to choose the order in which they will sit next to each other. The leftmost girl can choose seat nrs 1 till 6, so there are 6*4! ways in total to get the girls seated. There are 5! ways to get the boys seated, making the total number of ways to get the girls and boys seated such that all four girls are seated next to each other equal to 5!*6*4! = 4!*6!
In total there are thus 5!*6! + 4!*6! = 103680 ways to have at least 3 girls sitting next to each other.
To get precisely 3 girls next to each ther, you can choose the 3 girls and the order in which they will take their seats in 4! = 24 ways. There are 4! ways you can rearrange 4 girls, if we pick the first 3, you need to divide by the number of ways the girls we are not selcted can be rearranged, which is 1 in this case. Or you can say that you can choose the 3 girls in 4 ways, because this amounts to choosing the one who is going to be left out and then the 3 you choose can be rearranged in 3!= 6 ways and 3!*4 = 4!.
There are 7 seats available for each of these 24 possibilities. If we enumerate the seats from 1 to 9 counting from left to right, then the leftmost girl can take seat nrs 1 till 7. If she chooses seat nr. 1 or 7, then after the three girls are seated, out of the 6 remaining free seats, one seat will be excluded for the firth girl as she can't sit next to the other girls. There will then be 5 possible seats for her. So, in this case we have for each of the two possible seat choices for the leftmost girl, 4!*5 = 5! to get all the girls seated, so 2* 4!*5 = 2* 5! possibilities in total.
If the leftmost girl chooses seat nr. 2,3,4,5,or 6, then there are only 4 possible seats for the fourth girl. so 4!*4 ways to get the girls seated for each choice the leftmost girl makes, therefore 5*4*4! = 4*5! possibilities in total for these seat choices.
The total number of ways to get precisely 3 girls seated next to each other is thus 4*5! + 2*5! = 6*5! = 6!
Then there are 5! ways to get the five boys seated in the remaining seats, so there are 5!*6! ways to get the boys and girls seated such that there are precisely 3 girls sitting next to each other.
Then we need to consider the number of ways one can seat all the four grls next to each other. There are 4! ways to choose the order in which they will sit next to each other. The leftmost girl can choose seat nrs 1 till 6, so there are 6*4! ways in total to get the girls seated. There are 5! ways to get the boys seated, making the total number of ways to get the girls and boys seated such that all four girls are seated next to each other equal to 5!*6*4! = 4!*6!
In total there are thus 5!*6! + 4!*6! = 103680 ways to have at least 3 girls sitting next to each other.
Thank you , Count Iblis.
Count Ibis, you are awesome! I loved your explanation too.
1 answer