Solution: 2014 Winter Final - 4

Author: Michiel Smid

Question

Consider a group of 8 people, consisting of Simon, John, and 6 other people. How many ways are there to arrange these 8 people on a horizontal line such that Simon and John are standing next to each other? (The order on the line matters and Simon is either to the left or to the right of John.)

(a)

$2 \cdot 8 \cdot 6!$

(b)

$8 \cdot 6!$

(c)

$7 \cdot 6!$

(d)

$2 \cdot 7 \cdot 6!$

Solution

Since we want to keep Simon and John together, we can treat them as a single entity.

This reduces the problem to arranging 7 entities (Simon and John as one entity and the other 6 people) on a line.

There are 2 possible combinations with Simon on the left and John on the right, or Simon on the right and John on the left

The first entity (Simon and John) has 7 possible positions to choose from

The second entity has 6 possible positions to choose from

The third entity has 5 possible positions to choose from

…

The seventh entity has 1 possible position to choose from

The total number of ways to arrange the 8 people is $ 2 \times 7 \cdot 6! $