Solution: 2017 Fall Midterm - 2

Author: Michiel Smid

Question

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)

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

Solution

We know that Trumps can’t stand beside Kims, so the Supermans have to be in the middle.

It could look like TTSSKK or KKSSTT

So this doubles the number of possibilities.

We need to arrange the Supermans in their own section.

There are $ s! $ ways to do this.

We need to arrange the Trumps in their own section.

There are $ t! $ ways to do this.

We need to arrange the Kims in their own section.

There are $ k! $ ways to do this.

Thus, there are $ 2 \cdot s! \cdot t! \cdot k! $ ways to arrange the students.