Final answer:
The number of different rankings of the top 5 salespeople from a group of 39 is calculated using permutations. P(39, 5) = 39! / 34! simplifies to 39 × 38 × 37 × 36 × 35, which equals 69,090,840 possible rankings.
Step-by-step explanation:
To determine how many different rankings of the top 5 salespeople are possible from a group of 39, we can use the concept of permutations, which considers the order of selection. When we select 5 out of 39 without replacement and where the order matters, we calculate the number of permutations as follows:
P(n, k) = n! / (n - k)! where n is the total number of items to choose from, and k is the number of items to choose.
In this case, n = 39 salespeople and k = 5 top salespeople to rank. So the calculation would be:
P(39, 5) = 39! / (39 - 5)!
This simplifies to:
P(39, 5) = 39! / 34!
The factorial symbol (!) represents the product of all positive integers up to that number. Therefore:
39! = 39 × 38 × 37 × 36 × 35 × 34!
To find the number of possible rankings, we cancel out the 34! from both the numerator and denominator and multiply the remaining numbers:
39 × 38 × 37 × 36 × 35
By completing the multiplication, we find that there are a total of 69,090,840 possible rankings of the top 5 salespeople from 39.