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

  • Answer:

    968 ways

    Step-by-step 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!}{(n-r)!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.