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?
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
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.
Thank you , Count Iblis.
Count Ibis, you are awesome! I loved your explanation too.
To find the number of ways the boys and girls can be seated such that at least 3 girls are all next to each other, we can use the principle of complementary counting.
First, let's find the total number of ways to arrange the 5 boys and 4 girls in a row of 9 chairs without any restrictions.
The first chair can be occupied by any of the 9 people, so there are 9 choices.
After placing someone in the first chair, there are 8 remaining people to choose from for the second chair.
Continuing this process, the number of choices for each subsequent chair decreases by 1 until there are no more people left to sit.
Therefore, the total number of ways to arrange the boys and girls without any restrictions is 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880.
Now, let's count the number of ways that the girls can be seated such that at least 3 of them are all next to each other.
We can treat the group of at least 3 girls sitting together as a single entity. We have 4 girls, so we can choose a group of size 3 and leave the remaining girl ungrouped in 4 different ways.
After treating the group of girls as a single entity, we have a total of 6 entities (1 boys group, the girls group, and 4 individual boys) to arrange.
The entities can be arranged in 6! = 720 different ways.
Within the girls' entity, the 3 girls can also be arranged among themselves in 3! = 6 different ways.
Therefore, the total number of ways to arrange the boys and girls such that at least 3 girls are all next to each other is 4 * 720 * 6 = 17,280.
Finally, to find the number of ways to arrange the boys and girls in a row of 9 chairs such that at least 3 girls are all next to each other, we subtract the number of ways with restrictions from the total number of ways:
Total number of ways - Number of ways with restrictions = 362,880 - 17,280 = 345,600.
Therefore, there are 345,600 ways to seat the boys and girls in a row of 9 chairs such that at least 3 girls are all next to each other.