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How many ways are there to answer a 10-question true/false test, where at least 3 of the questions have been answered with a false?

 

I got 14 but that is incorrect.

Guest Jul 18, 2018
 #1
avatar+1139 
+1

Solution:

There are (10C3) 120 ways to choose where to place the three false responses. This leaves (27) 128 ways to respond to the remaining questions with true or false

------------------

How many ways are there to answer a 10-question true/false test, where at least 3 of the questions have been answered with a false?

There are (10C3) 120 ways to choose where to place the three false responses. This leaves (27) 128 ways to respond to the remaining questions with true or false

 

 (10C3)   = 120   *       128                   15360

 This continues with

(10C4)   = 210    *         64                   13440

(10C5)   = 252    *         32                     8064

(10C6)   = 210    *         16                     3360

(10C7)   = 120    *           8                       960

(10C8)     = 45    *           4                       180

(10C9)     = 10    *           2                         20

(10C10)     = 1    *           1                           1

 

Sums        968             256                 35185

Analysis:  There are 35185 ways to answer 10 question T/F with 3 or more pre-answered questions. Total includes multiple redundancies. At least 64 permutations will not exist in the totals.

 

 

GA

GingerAle  Jul 18, 2018
edited by GingerAle  Jul 18, 2018
edited by GingerAle  Jul 20, 2018
 #2
avatar
+1

Here's the answer: we can find the total number of combinations (without any restrictions) and then subtract the number of combinations that have less than 3 "false" answers.

 

the number of combinations without any restrictions is 210 (because there are 10 questions and each question can be answered with "true" or "false", meaning that there are 2 possible answers for each question).

 

the number of combinations with less that 3 questions answered with "false" is:

 

(number of combinations with exactly 2 questions answered with "false")+(number of combinations with exactly 1 question answered with "false")+(number of combinations with no questions answered with "false")= (10 choose 2)+(10 choose 1)+10 choose 0) (because we have to choose the questions that will be answered with "false")

 

(10 choose 2)+(10 choose 1)+10 choose 0)=56, so all we have to do to find the number of combinations that follow the restriction is to subtract 56 from 210=1024: 1024-56=968. and that's the answer to the question.

Guest Jul 18, 2018
edited by Guest  Jul 18, 2018
 #3
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0

Ginger has a wrong answer. That doesn’t happen very often. I wonder how she will respond.

 

Ginger says I’m an armchair mathematician. This is like an armchair quarterback except for mathematics.  I’m not very good at solving math problems but I usually recognize a correct answer and a well-presented solution when I see it.  Likewise, I usually recognize an incorrect answer and a poorly presented solution when I see it.

 

I didn’t notice Ginger’s answer was wrong until I saw this one. But I did notice this one is wrong. It’s not just wrong it’s fu*ked up!

Guest Jul 19, 2018
edited by Guest  Jul 19, 2018
 #4
avatar+1139 
0

Ginger speaks: Yep I screwed up. I neglected the caveat of “at least” in the question.

 

Ginger Troll speaks: I didn’t post a wrong answer. I correctly answered the wrong question. A home run on an error is still a home run. laugh

 

GA

GingerAle  Jul 19, 2018
 #6
avatar+1139 
0

Pending review....

GingerAle  Jul 19, 2018
edited by GingerAle  Jul 19, 2018
edited by GingerAle  Jul 20, 2018
 #7
avatar
0

The chimp has gone mad, a total basket case.

Guest Jul 20, 2018
edited by Guest  Jul 20, 2018

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