Question: 2017 Fall Midterm - 2

Author: Michiel Smid

Let $s \geq 1$, $t \geq 1$, and $k \geq 1$ be integers. The Carleton Computer Science Society is organizing their annual Halloween party. At this party, there are

$s$ students who are dressed up as Superman,
$t$ students who are dressed up as Donald Trump,
$k$ students who are dressed up as Kim Jong Un.

These $s+t+k$ students are arranged on a line, such that all Supermen are standing next to each other, all Trumps are standing next to each other, all Kims are standing next to each other, and no Trump is standing next to any Kim. How many such arrangements are there?

(a)

$2 \cdot s! \cdot t! \cdot k!$

(b)

$s! \cdot t! \cdot k!$

(c)

$\left. (s+t+k)! \middle/ (2 \cdot s! \cdot t! \cdot k!) \right.$

(d)

$\left. (s+t+k)! \middle/ (s! \cdot t! \cdot k!) \right.$