To find out how many ways 10 out of 15 people can take an elevator, we use the combination formula C(n, k) = n! / (k!(n-k)!), which gives us 3003 different combinations.
The question asks us to count the number of ways that the first group of people can take the elevator, given that there are 15 people in total and the elevator can only carry 10 people at a time.
This is a problem of combinations where we need to select 10 people out of 15 without considering the order.
To calculate the number of combinations, we can use the combination formula which is defined as:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of items, k is the number of items to choose, n! denotes the factorial of n, and (n-k)! denotes the factorial of n-k.
In our case, n is 15 and k is 10. Therefore:
C(15, 10) = 15! / (10!(15-10)!) = 15! / (10!5!) = 3003
So, there are 3003 different ways in which the first group of 10 people can be selected to take the elevator.