For the questions, a club has 10 members.
a. find the number of different slates of 3 officers (list of president, vice-president, and treasurer) that the club could have for officers.
b. find the number of different slates of 4 officers (list of president, vice-president, secretary, and treasurer) that the club could have for officers.

How many slates = How many combinations of people/jobs.

a. We have 10 different choices for our president.
Then, once we've chose a president, that leaves 9 for our VP.
And then 8 for our treasurer.
(the decrease is necessary b/c we can't have one person fill two positions)

10 × 9 × 8 = 720

b. In a similar format to part a...

10 × 9 × 8 × 7 = 5040