Question

At UC San Diego, incoming students are each randomly assigned to join one of the six colleges: Revelle, Muir, Marshall, Warren, Roosevelt, and Sixth. Suppose there are 24 incoming students, including Winona, Xanthippe, and Zelda. It cannot be assumed that all colleges will be assigned four students; since each student’s assignment is random, some colleges may be assigned more students than others.
Please leave your answers as unsimplified expressions with factorials, exponents, permutations, combinations, etc.
a) All 24 incoming students get called up one at a time to be assigned to a college. How many possible orders of all 24 students have Winona as the first student called?
b) How many possible orders of all 24 students have Zelda, Xanthippe, and Winona (in any order) as the first three students called up to wear the sorting hat?
c) How many ways are there to assign all 24 students to colleges such that Zelda, Xanthippe, and Winona all get assigned to Muir? Here, and in what follows, the order in which the students are assigned does not matter.
d) How many ways are there to assign all 24 students to colleges such that Zelda, Xanthippe, and Winona all get assigned to the same college?
e) How many ways are there to assign all 24 students to colleges such that Zelda, Xanthippe, and Winona are assigned to different colleges?
f) How many ways are there to assign all 24 students to colleges such that exactly 4 students get assigned to Warren?
g) How many ways are there to assign all 24 students to colleges if each of the six colleges has room for only four incoming students?
h) How many ways are there to assign all 24 students to colleges if nobody is assigned to the same colleges as the person called up just before them?
i) What is the probability that Winona, Xanthippe, and Zelda are all assigned to Muir?
j) Suppose again that the students are called up one at a time. In how many possible orders of all 24
students does Zelda get called up some time before Xanthippe? Simplify your answer as much as possible, but let factorials remain unsimplified.

Answer 1

(a)

Winona is called in the first and remaining 23 students were called in 23! ways.

So, number of possible orders is 23!.

(b)

Zelda, Xanthippe and Winona can be called in first three in 3! ways and remaining 21 students in 21! ways.

So, number of possible orders is 3!*21!.

(c)

Zelda, Xanthippe and Winona are assigned in 1 college Muir. All of other 21 students can be assigned to any of 6 colleges each in 6²¹ ways.

So, number of possible orders is 6^21.

(d)

Zelda, Xanthippe and Winona can be assigned to same college in 6 ways and remaining 21 students can be assigned to any of 6 colleges each in 6²¹ ways.

So, number of possible orders is 6^22.

(e)

Zelda, Xanthippe and Winona can be assigned to different colleges in 6*5*4 ways and remaining 21 students can be assigned to any of 6 colleges each in 6²¹ ways.

So, number of possible orders is 20*6^22.

(f)

Exactly 4 students can be assigned in Warren in ²⁴C₄ ways and remaining 20 students can be assigned to other colleges in 5²⁰ ways.

So, number of possible orders is ²⁴C₄*5²⁰.