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**+1 **

Solution:

There are (10C3) 120 ways to choose where to place the three *false* responses. This leaves (2^{7}) 128 ways to respond to the remaining questions with *true *or* false*.

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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?}

There are (10C3) 120 ways to choose where to place the three *false* responses. This leaves (2^{7}) 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

#2**+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 2^{10} (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 2^{10}=1024: 1024-56=968. and that's the answer to the question.

Guest Jul 18, 2018

edited by
Guest
Jul 18, 2018

#3**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