rajatvmittal wrote:A computer game has five difficulty levels. In each level you can choose among four different scenarios except for the first level, where you can choose among three scenarios only. How many different games are possible?
Note: The question is somewhat ambiguous. I'm assuming that the scenarios for each level are considered different. For example, completing scenario B at level 2 is different from completing scenario B at level 3.
Take the task of playing a game (i.e., complete all 5 levels) and break it into stages.
Stage 1: Complete level 1
There are 3 scenarios to choose from, so this stage can be accomplished in
3 ways.
Stage 2: Complete level 2
There are 4 scenarios to choose from, so this stage can be accomplished in
4 ways.
Stage 3: Complete level 3
There are 4 scenarios to choose from, so this stage can be accomplished in
4 ways.
Stage 4: Complete level 4
There are 4 scenarios to choose from, so this stage can be accomplished in
4 ways.
Stage 5: Complete level 5
There are 4 scenarios to choose from, so this stage can be accomplished in
4 ways.
By the Fundamental Counting Principle (FCP) we can complete all 5 stages (and thus complete all 5 levels) in
(3)(4)(4)(4)(4) ways ([spoiler]= 768 ways[/spoiler])
Cheers,
Brent
Aside: For more information about the FCP, we have a free video on the subject:
https://www.gmatprepnow.com/module/gmat-counting?id=775
