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Solution: 2018 Fall Midterm - 15

Author: Michiel Smid

Question

The Carleton Computer Science Society is organizing their annual Halloween party. At this party,
  • one student is dressed up as Donald Trump,
  • one student is dressed up as Kim Jong Un,
  • the remaining 57 students are dressed up as Kim Kardashian.
These students are arranged, uniformly at random, on a line.
Define the event,
  • A = "Donald Trump is standing next to Kim Jong Un".
What is $\Pr(A)$?
(a)
1/59
(b)
2/58
(c)
1/58
(d)
2/59

Solution

When looking for possible combinations of Trump and Kim standing together, we can treat them as a single entity in terms of position.

There are 2 possible ways to arrange Trump and Kim together.

Trump on left and Kim on right OR Kim on left and Trump on right: 2

When we do view them as a single entity together, we are left with 58 entities.

The third person has 57 possible positions to stand.

The fourth person has 56 possible positions to stand.

So on and so forth until the 59th person has 1 possible position to stand.

Thus, there are $ 2 \cdot 58 \cdot 57! $ ways to arrange the students.

There are a total of $ 59! $ ways to arrange the students if we didn’t care about combination restrictions.

Thus, Pr$ (A) = \frac{2 \cdot 58 \cdot 57!}{59!} = \frac{2}{59} $