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

Answer:

Option D - $\frac{2}{7}$.

Step-by-step 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 $^8C_4$

$^8C_4=\frac{8!}{4!(8-4)!}$

$^8C_4=\frac{8\times 7\times 6\times 5\times 4!}{4!\times 4\times 3\times 2}$

$^8C_4=70$

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 $^6C_3$

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

$^6C_3=\frac{6\times 5\times 4\times 3!}{3!\times 3\times 2}$

$^6C_3=20$

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

$P=\frac{^6C_3}{^8C_4}$

$P=\frac{20}{70}$

$P=\frac{2}{7}$

The probability that Andrew will be among the 4 volunteers selected and Karen will not is $\frac{2}{7}$.

Therefore, option D is correct.