Question

11. Jillian needs to make a group of 5 people. She has 10 Democrats, 13 Republicans, and 7 Independents to choose from. (a) What is the probability that the group of 5 people will have 2 Democrats, 2 Republicans, and 1 Independent

Answer 1

Answer:

0.1724 or 17.24%

Step-by-step explanation:

There are  

10+13+7 = 30 people

so there are C(30,5) combinations of 30 elements taken 5 at a time ways to select groups of 5 people.

$C(30,5)=\displaystyle\binom{30}{5}=\frac{30!}{5!25!}=142,506$

There are C(10,2) combinations of 10 elements taken 2 at a time ways to select the 2 Democrats

$C(10,2)=\displaystyle\binom{10}{2}=\frac{10!}{2!8!}=45$

There are C(13,2) combinations of 13 elements taken 2 at a time ways to select the 2 Republicans

$C(13,2)=\displaystyle\binom{13}{2}=\frac{13!}{2!11!}=78$

There are C(7,1) combinations of 7 elements taken 1 at a time ways to select the Independent

$C(7,1)=\displaystyle\binom{7}{1}=\frac{7!}{1!6!}=7$

By the Fundamental Principle of Counting, there are  

45*78*7 = 24,570

ways of making the group of 5 people with 2 Democrats, 2 Republicans and 1 independent, and the probability of forming the group is

$\displaystyle\frac{24,570}{142,506}=0.1724$ or 17.24%