Question

1. A state legislature consists of 7 Democrats, 4 Republicans, and 8 Independents. How many ways are there to choose:

a. A committee of 5 legislators?

b. A speaker, a vice-speaker, and a sergeant at arms?

c. A committee of 6 legislators where 2 are Republicans, and 2 are Democrats, and 2 are Independents?

d. A committee of 5 legislators, with no Independents?

2. A "2 for $20" meal deal at a local restaurant allows you to pick 1 appetizer from 5 options, 2 entrees from 6 choices, and 1 desert from 2 choice(s).

How many different meals are possible?

3. Candy! 3 Reese's, 7 Rolos, and 15 Kisses are are mixed up randomly in a dish. You reach in without looking and randomly grab 5 candies. Find:

a. P(all 5 are Rolos):

b. P(all 5 are Kisses):

c. P(one candy is Reese's, the rest are Rolos):

d. P(two candies are Rolos, the rest are not):

e. P(no candies are Rolos):

f. P(at least one candy is a Rolo):

Answer 1

Question 1:

a) Number of ways to chose 5 legislators is computed here as:
= Number of ways to select 5 people from (7 + 4 + 8) = 19 people

b) Total number of ways to choose A speaker, a vice-speaker, and a sergeant at arms is computed here as:

= Total permutation of 19 people taken 3 at a time

(Note that permutations are also important here as we are choosing for different positions )

c) Total number of ways to choose A committee of 6 legislators where 2 are Republicans, and 2 are Democrats, and 2 are Independents is computed here as:

= Number of ways to select 2 democrats from 7 democrats * Number of ways to chose 2 republicans from 4 republicans * Number of ways to select 2 independents from 8 independents

d) Number of ways to select A committee of 5 legislators, with no Independents is computed here as:

= Number of ways to select 5 people from (7 + 4) = 11 non independents