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?
Mathematics
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Question
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?
1 Answer

1. User Answers kalyani62
Answer:
968 ways
Stepbystep explanation:
This is a question of permutation and combination.
Each equation can have two different answers.
Thus the total number of cases will be (for 10 questions) :
[tex]2*2*2*2*.....10times=2^{10}[/tex] cases.
Now to find the number of ways to at least answer 3 questions False will be total minus the number of question with at most 2 False answers.
 Number of ways in which no answer is False : 1 ( all are true )
 Number of ways in which ONLY one answer is False : [tex]10_C_1[/tex] where [tex]n_C_r=\frac{n!}{(nr)!r!}[/tex]
 Number of ways in which ONLY two answers are False :[tex]10_C_2[/tex]
Total ways (at most 2 answers false) = [tex]1+10_C_1+10_C_2[/tex] ;
∴
The number of ways in which at least 3 have False as the answer is :
[tex]2^{10}(1+10_C_1+10_C_2)\\=968[/tex] WAYS.