We hope everyone had a happy Halloween! Yesterday we asked our friends on our Manhattan GMAT Facebook page to attempt this spooky Halloween Challenge Problem. As promised, today we are sharing the answer and explanation to the problem:
This question is not as complicated as it may initially seem. The trick is to recognize a recurring pattern in the assignment of the ghouls.
First, we have five ghouls (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 ghouls 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.