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3. Hogwarts Counting Practice Please give your answers as an exact unsimplified formula, and as a probability rounded to four decimal places At Hogwarts School of Witchcraft and Wizardry, incoming students are each as- signed to join one of the four houses: Gryffindor, Hufflepuff, Ravenclaw, and Sl there are 24 incoming students, including Harry, Ron, and ytherin. Suppose Hermione. All 24 incoming students get called up one at a time to wear a sorting hat that will select which house they will join. (a) How many possible orders of all 24 students have Hermione as the first student called, Harry as the second and Ron as the third? (b) How many possible orders of all 24 students have Harry, Ron, and Hermione (in any order) as the last three students called up to wear the sorting hat? (c) How many ways are there to assign all 24 students to houses such that Harry, Ron, and Hermione all get assigned to Gryffindor? (d) How many ways are there to assign all 24 students to houses such that exactly 6 students get assigned to Gryffindor? (e) How many ways are there to assign all 24 students to houses if nobody is assigned to the same house as the last two persons called up to the sorting hat just before them? In other words, the student in position n should be assigned to a different house than the ones assigned to the students in position n -1 and (f) If the students are divided in the way described in (e), can you calculate the maximum number of students that can be assigned to Slytherin?

Answer 1

(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 $24-3=21$ ways, 5th position in $24-3-1=20$ ways and so on. So the remaining positions can be fill in $21.20.19....3.2.1=21!$ ways. Hence the ordering can be done in $21!$ 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 $21!$ many ways, and for the last three there are $3!$ many choice. Hence the total way we can do this in $21!\times3!$ 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 $1^3\times 4^{21}=4^{21}$ .

(d) We can choose 6 students from 24 in $\binom{24}{6}$ 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 $4^2\times 3\times 2^{21}$

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 $4\times 3\times 2^{22}$

(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 $2+7=9$ 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!