Problem Solving

Most Manhattan GMAT students are trying to break the 700 barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you'll WANT to see, when you are working at that level. Try to solve this 700+ level problem (I'll post the solution next Monday).

Question - Guard Patrol:
The organizers of a week-long fair have hired exactly five 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?

A) 9
B) 7
C) 5
D) 3
E) 2

After first day on any day there are three guards to choose from hence the number of pairs are 3C2= 3
Am i right?

Answer
This question is not as complicated as it may initially seem. The trick is to recognize a recurring pattern in the assignment of the guards.

First, we have five guards (let's call them a, b, c, d, and e) and we have to break them down into pairs. So how many pairs are possible in a group of five distinct entities?

We could use the combinations formula: ,

where n is the number of items you are selecting from (the pool) and k is the number of items you are selecting (the subgroup).

Here we would get .

So there are 10 different pairs in a group of 5 individuals.

However, in this particular case, it is actually more helpful to write them out (since there are only 5 guards and 10 pairs, it is not so onerous): ab, ac, ad, ae, bc, bd, be, cd, ce, de. Now, on the first night (Monday), any one of the ten pairs may be assigned, since no one has worked yet. Let's say that pair ab is assigned to work the first night. That means no pair containing either a or b may be assigned on Tuesday night. That rules out 7 of the 10 pairs, leaving only cd, ce, and de available for assignment. If, say, cd were assigned on Tuesday, then on Wednesday no pair containing either c or d could be assigned. This leaves only 3 pairs available for Wednesday: ab, ae, and be.

At this point the savvy test taker will realize that on any given night after the first, including Saturday, only 3 pairs will be available for assignment.

Those test takers who are really on the ball may have realized right away that the assignment of any two guards on any night necessarily rules out 7 of the 10 pairs for the next night, leaving only 3 pairs available on all nights after Monday.

The correct answer is Choice D; 3 different pairs will be available to patrol the grounds on Saturday night.

Question - Guard Patrol:
The organizers of a week-long fair have hired exactly five 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?

A) 9
B) 7
C) 5
D) 3
E) 2

Let guards be A,B,C,D and E
As no guard can be assigned consecutive nights, Guards on Friday can't be assigned.
2 guards will be assigned on Friday. Thus remaining = 3 (=5-2)
out of these 3, we have to select 2 guards for duty on Saturday
3C2 = 3
IMO D

3c2 is the answer