9. a band director wants to form a committee of 4 parents from a list of 45 band parents.
a. how many different groups of 4 parents can the band director select?
b. how many different ways can the band director select 4 parents to serve in the band parents’ association as president, vice president, treasurer, and secretary?
c. explain the difference between parts (a) and (b) in terms of how you decided to solve each part.

Part (A): The number of ways to select 12 jurors are 86493225.

Part (B): The number of ways to select 2 alternates are 306.

Part (C): Part (a) is an example of combination  and Part (a) is an example of permutation

Step-by-step explanation:

Consider the provided information.

A band director wants to form a committee of 4 parents from a list of 45 band parents.

Part (A):  How many different groups of 4 parents can the band director select?

Since order doesn't matter so we will use combination.

Use the formula for combination

Substitute n=45 and r=4 in above formula.

Hence, the number of ways are 148995.

Part (B) How many different ways can the band director select 4 parents to serve in the band parents’ association as president, vice president, treasurer, and secretary?

Since the order matter so we will use permutation:

The formula for permutation is:  

Substitute n=45 and r=4 in above formula.

Hence, the number of ways are 3575880.

Part (C) . Explain the difference between parts (a) and (b) in terms of how you decided to solve each part.

Part (A) is an example of number of combination where order doesn't matter.

Part (B) is an example of permutation where order matter.

picture below