Final answer:
There are 3024 ways to assign four distinct positions for a relay race from a team of nine runners, calculated using the permutation formula P(n, k) = n! / (n-k)!, where n is the number of items and k is the number of positions to fill.
Step-by-step explanation:
A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}. As you can see, there are no other ways to arrange the elements of set A.
In permutation, the elements should be arranged in a particular order whereas in combination the order of elements does not matter.
Basically Permutation is an arrangement of objects in a particular way or order. While dealing with permutation one should concern about the selection as well as arrangement. In Short, ordering is very much essential in permutations. In other words, the permutation is considered as an ordered combination.
The question is asking to calculate in how many ways four distinct positions for a relay race can be filled by nine runners. This is a problem of permutations since the order of the positions matters. To solve this, you can use the permutation formula:
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.
For this particular question, you have nine runners (n=9) and you want to choose four of them for distinct positions (k=4).
The calculation is then:
P(9, 4) = 9! / (9-4)! = 9! / 5! = 9 × 8 × 7 × 6 = 3024 ways
So there are 3024 ways to assign four distinct positions for a relay race from a team of nine runners.