Question

In an examination a candidate is required to pass at least four different subjects out of six. Then the number of ways in which he can fail is (a)12
(b)16
(c)24
(d)42

Answer 1

I m not sure about this question

Answer 2

Given details:

Total number of subjects = 6

To pass in the examination, a cadidate has to pass in at least 4 subjects.

To find:

Number of ways to fail.

To fail this examination a candidate has to fail in

any 3 subjects or 4 subjects or 5 or all 6 subjects.

To choose r objects out of n objects, the formula of combination is used.

$^nC_r = \frac{n!}{(n-r)!(r)!}$

So, the ways to fail in 3 subjects are (choosing any 3 subjects out of 6)

$^6C_3 = \frac{6!}{(6-3)!(3)!} = \frac{6!}{3!3!}= 20$

The ways to fail in 4 subjects are (choosing any 4 subjects out of 6)

$^6C_4 = \frac{6!}{(6-4)!(4)!} = \frac{6!}{2!4!}= 15$

The ways to fail in 5 subjects are (choosing any 5 subjects out of 6)

$^6C_5 = \frac{6!}{(6-5)!(5)!} = \frac{6!}{1!5!}= 6$

The ways to fail in 6 subjects are (choosing 6 subjects out of 6)

$^6C_6 = \frac{6!}{(6-6)!(6)!} = \frac{6!}{0!6!}= 1$

$^6C_3 + ^6C_4 + ^6C_5 + ^6C_6 = 20 + 15 + 6 + 1 = 42$

$\textbf{\Large The number of ways by which he can fail =42}$