Question

an archaeology club has 45 members. how many different ways can the club select a president, vice president, treasurer, and secretary?

there are different slates of candidates possible.
(simplify your answer.)

Explanation:

Identify the counting scenario

Using the Permutations vs Combinations knowledge point
Since the selected members fill distinct, ordered roles (president, vice president, treasurer, and secretary), the order of selection matters. This is a permutation problem where we select \(r = 4\) distinct positions from a pool of \(n = 45\) members.

Apply the permutation formula

Using the Permutations vs Combinations knowledge point

$$ P(n, r) = \frac{n!}{(n-r)!} $$

$$ P(45, 4) = \frac{45!}{(45-4)!} = \frac{45!}{41!} $$

Calculate the product

Using the Fundamental Counting Principle knowledge point

$$ P(45, 4) = 45 \times 44 \times 43 \times 42 $$

$$ 45 \times 44 = 1980 $$

$$ 1980 \times 43 = 85140 $$

$$ 85140 \times 42 = 3575880 $$

Answer:

There are <blank>3,575,880</blank> different slates of candidates possible.