Find the number of ways the top four can finish in a race of nine people.
Assuming order matters, i.e.
A,B,C,D is counted differently as B,A,C,D.
There are 9 choices for the first place, 8 places for the second, 7 places for the third, and 6 places for the fourth. Applying the multiplication rule, there are 9*8*7*6 ways, or
P(9,4) [permutations]
=9!/(9-4)!
If order does not matter, the above must be divided by 4! combinations among the top four, or
number of ways
=C(9,4)
=9!/[(9-4]!4!]
To find the number of ways the top four can finish in a race of nine people, we need to use the concept of permutations.
In a race of nine people, the first-place finisher can be any of the nine people. For the second-place finisher, there are eight remaining choices, since one person has already finished in first place. Similarly, for the third-place finisher, there are seven remaining choices, and for the fourth-place finisher, there are six remaining choices.
Therefore, the number of ways the top four can finish in a race of nine people is calculated as follows:
Number of ways = 9 x 8 x 7 x 6 = 3024
So, there are 3024 different ways the top four can finish the race out of nine people.
To find the number of ways the top four can finish in a race of nine people, we can use the concept of permutations.
Permutations are used to calculate the number of arrangements of objects. In this case, we want to find the number of ways the top four can finish, which means we are interested in the order in which they finish.
To solve this problem, we can use the permutation formula.
The permutation formula is given by:
P(n, r) = n! / (n - r)!
Where P(n, r) represents the number of permutations of n objects taken r at a time, and ! denotes the factorial of a number.
In this case, we want to find the number of ways the top four can finish, so we have four objects (the top four) and we are arranging them in a certain order. Therefore, n = 9 (total number of people) and r = 4 (top four).
Using the permutation formula, we can calculate the number of ways the top four can finish:
P(9, 4) = 9! / (9 - 4)!
= 9! / 5!
Calculating the factorials:
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
Now, we can substitute these values into the formula and simplify the expression:
P(9, 4) = (9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (5 x 4 x 3 x 2 x 1)
= (9 x 8 x 7 x 6) / (4 x 3 x 2 x 1)
Calculating:
P(9, 4) = 3024
Therefore, there are 3024 different ways the top four can finish in a race of nine people.