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3 girls A B C and 9 boys are to be lined up in a row .In how many ways can this be done if B says she wants to be between A and C.Note that if the boys are shown w...

Question

3 girls A B C and 9 boys are to be lined up in a row .In how many ways can this be done if B says she wants to be between A and C.Note that if the boys are shown with numbers 1,2,3, .......9,then the following orderings are acceptable: A, 1,2,3,.........,9, B,C or C,1,......,8,9,A, while B, A C ,1,.......,9 is not acceptable.

3 girls A B C and 9 boys are to be lined up in a row .In how many ways can this be done if B says she wants to be between A and C.Note that if the boys are shown with numbers 1,2,3, .......9,then the following orderings are acceptable: A, 1,2,3,.........,9, B,C or C,1,......,8,9,A, while B, A C ,1,.......,9 is not acceptable.



Answers

In how many ways can four girls and two boys be arranged in a row if a) the boys are on each end of the row? b) the boys must be together? c) the boys must be together in the middle of the row?

In this question, we have to determine the number of ways. We can arrange four girls and two boys in a row in the following different scenarios. So in the first one, the boys are on each end of the road. So we know we have six people total, so we can make six slots for them. And we know that each boy must be on the end, so we have boy one and boy too. So we know that there's two ways we can pick which boy is on the end, since we can either pick one of the two boys in the first end and then only one other boy left to fill the second end. Next we have four different girls, we have to fill in the middle, so we can pick four girls for the first one, then we only have three left, then two and finally won. So you can see this amounts to four factorial times two factorial, which is just the four times three times two times one, times two times one. And when we multiply this out we'll get that. There are 48 ways to arrange the boys and girls. If we're keeping the boys on the end. Now in the second scenario, the boys must be together but we're not told where in this scenario. So once again we have our six lots and now we have to see the number of ways we can keep the boys together. So we could have boy one and boy to in the first two slots. They could also be in the second to or in slots three and four and four and five, and then finally five and six. So we can see there's five boys for the boys to be together. Next we know that since we have two boys, we can pick either one of them. So we have five ways, times two factorial ways to pick which boy is first and which boy is second. Finally similar to the first question, we still only have four factorial ways to arrange the girls and all the other slots. So then our final answer we get five times two factorial times four factorial. So now we can just simplify and solve this and we'll get 240 ways as our final answer. Now in the final problem, we have to put the boys together, but they have to be in the middle. So we have 123456 slots again and our boys have to be in the middle. So again, there's only two ways to pick which boy goes in which slot. And finally we once again have four factorial different ways to arrange the girls. So once again we get four factorial times two factorial which gets us 48 different ways to arrange them.

In this problem We have three boys and five girls. So eight students in total and they have to set Now the sitting pattern has to look like girl and a dash ago. Then blank girl in a blank, a girl, a blank, a girl and blanks on either side. This is how the sitting pattern should look like. And out of these six empty spaces, three of the spaces have to be occupied by boys. So six spaces three to be occupied. Bye boys. Now these five girls Fosters. The number of ways that the five girls can sit in any of the five places will be given by five B five. This is nothing but five factorial which is 120 earned. Out of these six places, boy, three boys have to sit. So the number of ways that three boys can sit in six places is 63, which is nothing but Six factory by 6 -3, which is three factorial. This comes us three factorial times 4 to 5 to six by three factories. Let's get canceled. This is nothing. But for and if I was 20 in two sixes, warned windy. So total number of ways in which sitting arrangement can be danced that no two boys are together is 1 20 to 1 20. This is 144 double zero.

In this problem, we have been asked to determine in how many ways three boys and four girls can sit in a rule that the boys and girls must alternate. Now there are three boys and four girls, so there are a total of seven spaces, and if the boys and girls must alternate, the note that the girls must Sit in the girls and boys must sit in this manner where G represents a girl and we represents a boy. Note that there are a total of 1234 girls and 123 boys. And in this arrangement they have sit in an alternate way. The boys and girls have alternated. So know that there are a total of 1234 places in which the four girls can sit. So four girls must arrange themselves in these four places and that can be done in four factorial number of weeks. Similarly, the boys have three places where they can sit and there are three of them. So the three boys can arrange themselves in these three places in a total of three factorial ways. So the total number of ways that this can be done can be obtained by the multiplication rules. So we multiply these two numbers. So four factorial is four times three times two times 13 factorial is three times two times one, and the value of this will be equal to 144. Hence the total number of ways that three boys and four girls can sit in a row. The boys and girls must alternate is equal to 144.

In this question, it is told that there are 15 men out of them three or some special. Or we can say that particular men and well you are the normal other men's. And it is told that we have to select nine men out of this 15 available. And in the first case it is told that we have to include three particular. It means that if we are including three particular, it means that out of that well, Normal, we have to take six members, six selection put on the remaining well. And this is done by the formula That is 12, say 600's value is 924. This is the answer of the first question in the Southern Ocean. It is told that we are doing exclude the three particular men's. It means that now we have to select nine from the remaining 12 members. It means that 12 C9 And 29 is 220. So 220 is the answer for the second question, 924 is the answer for the first question. Okay, thank you.


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