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#1 2006-12-24 00:00:15

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Yeses and Nos

A student randomly guesses each answer to a quiz consisting of 8 Yes/No questions. In how many different ways can he answer the 8 questions if he writes down more Yeses than Nos?
(Answer = 93)

I figured there were 3 scenarios. Yes = No, Yes > No and Yes < No. I also figured that Yes > No = Yes < No, as they are interchangeable. So:
For the scenario Yes = No, there are 70 different ways of answering:


So i figured:


But what is the total amount?
This question I think deals with several different concepts so I got really confused somewhere...

Am I right so far? Help please.

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#2 2006-12-24 01:45:15

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Yeses and Nos

Everything so far is right, and in fact, you've already done most of the work.

In total, there are 2^8 = 256 ways of answering all of the questions, because there are 2 ways of answering each question and 8 questions.

Your reasoning about Yes>No = No>Yes was correct, so we can put that into the formula you made.

Total number of Yes>No ways: (256-70)/2 = 186/2 = 93.


Why did the vector cross the road?
It wanted to be normal.

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#3 2006-12-24 06:21:03

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Yeses and Nos

Pretty good problem.
Let the n be the number of the questions.
In general we'll have two cases:
1. If n is odd, then we can't have (YES=NO) case. So in this case, the answer is 2^(n-1).
2. If n is even, we'll have for the number of (YES=NO) cases:


and the result will be:
.

Last edited by krassi_holmz (2006-12-24 06:21:30)


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