Solution: 2017 Winter Final - 11

Author: Michiel Smid

Question

Consider a group consisting of 7 girls and 6 boys. Elisa is one of the girls. How many ways are there to arrange these 13 people on a horizontal line such that Elisa has 2 neighbors, both of whom are girls? (The order on the line matters.)

(a)

$13 \cdot 6 \cdot 5 \cdot 10!$

(b)

$11 \cdot {6 \choose 2} \cdot 10!$

(c)

$12 \cdot 6 \cdot 5 \cdot 10!$

(d)

$11 \cdot 6 \cdot 5 \cdot 10!$

Solution

This means there are 13 positions

First, select 2 of the girls to be Elisa’s neighbors: $ \binom{6}{2} $

Since we’re putting 2 girls beside Elisa, girl 1 can be on the left and girl 2 can be on the right OR girl 1 can be on the right and girl 2 can be on the left: 2

As a single entity, the 3 girls can be placed, starting from position 0 to position 10: 11

The remaining 10 people can be placed in the remaining 10 positions: 10!

$ \binom{6}{2} \cdot 2 \cdot 11 \cdot 10! $

$ = \frac{6!}{2!4!} \cdot 2 \cdot 11 \cdot 10! $

$ = \frac{6!}{4!} \cdot 11 \cdot 10! $

$ = \frac{6 \cdot 5 \cdot 4!}{4!} \cdot 11 \cdot 10! $

$ = \frac{6 \cdot 5}{1} \cdot 11 \cdot 10! $

$ = 6 \cdot 5 \cdot 11 \cdot 10! $