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?

Answer 1

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) :

$2*2*2*2*.....10times=2^{10}$ 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 : $10_C_1$ where $n_C_r=\frac{n!}{(n-r)!r!}$

• Number of ways in which ONLY two answers are False :$10_C_2$

Total ways (at most 2 answers false) = $1+10_C_1+10_C_2$ ;

∴

The number of ways in which at least 3 have False as the answer is :

$2^{10}-(1+10_C_1+10_C_2)\\=968$ WAYS.