The organizers of a week-long fair have hired exactly seven security guards to patrol the fairgrounds at night for the duration of the event. Exactly two guards are assigned to patrol the grounds every night, with no guard assigned consecutive nights. If the fair begins on a Monday, how many different pairs of guards will be available to patrol the fairgrounds on the following Saturday night?
Choose the correct choice from the following options:
A) 6 B) 8 C) 10 D) 12 E) 14
For any given day, only the guards patrolling on the previous day won't be available. So, 2 guards who patrolled on Friday won't be available. We are thus left with 5 guards.   To choose 2 out of 5, we will have 5C2 = 10 different pairs.   The answer is 10.
<result>C</result>
The organizers of a week-long fair have hired exactly seven security guards to patrol the fairgrounds at night for the duration of the event. Exactly two guards are assigned to patrol the grounds every night, with no guard assigned consecutive nights. If the fair begins on a Monday, how many different pairs of guards will be available to patrol the fairgrounds on the following Saturday night?