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(a) Note that to get Hermione as first Harry as second and Ron as third, We need to fix those first three positions in exactly 1 way, and the forth position in ways, 5th position in ways and so on. So the remaining positions can be fill in ways. Hence the ordering can be done in ways.
(b) To get Hermione, Harry and Ron (in any order) in last 3, we need to fix the last three positions and the first 21 positions can be filled in all possible way. Now the discussion we made in part a, all possible way to fill up the first 21 positions is many ways, and for the last three there are many choice. Hence the total way we can do this in ways.
(c) Top get Hermione, Harry and Ron in Gryffindor , for Hermione, Harry and Ron, the number of choice for them to get in a house is 1 each, and for other students, the number of choice is 4 for each student. Hence the total way to happen Hermione, Harry and Ron in Gryffindor is .
(d) We can choose 6 students from 24 in ways, send them to Gryffindor and other to some other house.
(e) Note that for the first people the choice is 4, for the second people the choice is also 4, (note that the second people can be selected to the house same as the first people, as there is no two students before the second student), for the third student the choice can be 3 if the first and second student can be selected in the same house and 2 if the first and second student selected in the different house.
Case - 1 If the first and second student can be selected in the same house then for the third people the choice is 3, for the forth people the choice is 2, for the fifth people the choice is also 2, and for all other people the choice is 2, hence the total choice is
Case - 2 if the first and second student selected in the different houses. then the first people can have 4 choice, the second person can have 3 choices, and from the third to onwards the choice is 2. Hence the total choice
(f) Case - 1 If the first and second student can be selected in the same house, to get max number in Slytherin, we put first 2 in Slytherin, then third and fourth cann't have Slytherin, the fifth one can, again the 8th one can. So by this way we can have way.
Case - 2 if the first and second student selected in the different houses. To get max Slytherin we need to send the first student to Slytherin, then 2nd and third in other house, and fourth one in Slytherin and so one. So in this process we have 8 way.
Feel free to comment if you have any doubts. Cheers!
Will upvote/rate! Please show all work :) 3. Hogwarts Counting Practice Please give your answers as an exact unsimplifi...
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