You get to go to the horse races and see that nine horses are running in the fifth race. How many ways can the race end with first, second and third place winners?
9P3 = _____
there are 9 possible 1st place finishers (all the horses have a chance)
after the 1st place finisher, there are 8 remaining for possible 2nd place
after the 2nd place finisher, there are 7 remaining for possible 3rd place
9 * 8 * 7 = ?
To determine the number of ways the race can end with first, second, and third place winners, we can use the concept of permutations.
In this case, we need to find the number of permutations of 9 horses taken 3 at a time, because we want to select the first, second, and third place winners from the 9 horses in the race.
The formula to calculate permutations is given by:
P(n, r) = n! / (n-r)!
Where:
- n is the total number of objects (in this case, horses)
- r is the number of objects we are selecting (in this case, places: 3)
- ! denotes the factorial (the product of all positive integers less than or equal to n)
Using the formula, we can calculate the number of ways as:
P(9, 3) = 9! / (9-3)!
= 9! / 6!
= (9 * 8 * 7 * 6!) / 6!
= 9 * 8 * 7
= 504
Therefore, there are 504 different ways the race can end with first, second, and third place winners.