Question

From a group of 8 volunteers, including Andrew and Karen, 4 people are to be selected at random to organize a charity event. What is the probability that Andrew will be among the 4 volunteers selected and Karen will not?A. 3/7

B. 5/12
C. 27/70
D. 2/7
E. 9/35

Answer 1

Option D -
`(2)/(7)`.

Explanation:

Given : From a group of 8 volunteers, including Andrew and Karen, 4 people are to be selected at random to organize a charity event.

To find : What is the probability that Andrew will be among the 4 volunteers selected and Karen will not?

Solution :

Choosing 4 people out of 8 volunteers is
`^8C_4`

`^8C_4=(8!)/(4!(8-4)!)`

`^8C_4=(8* 7* 6* 5* 4!)/(4!* 4* 3* 2)`

`^8C_4=70`

Choosing a group of 4 with Andrew and no karein is given by,

One position is fixed by Andrew and Karein the number of volunteer left is 6.

Rest 3 volunteers is chosen from 6.

Choosing 3 people out of 6 volunteers is
`^6C_3`

`^6C_3=(6!)/(3!(6-3)!)`

`^6C_3=(6* 5* 4* 3!)/(3!* 3* 2)`

`^6C_3=20`

The probability that Andrew will be among the 4 volunteers selected and Karen will not is given by,

`P=(^6C_3)/(^8C_4)`

`P=(20)/(70)`

`P=(2)/(7)`

The probability that Andrew will be among the 4 volunteers selected and Karen will not is
`(2)/(7)`.

Therefore, option D is correct.