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

968 ways

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=(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.