Final answer:
There are 117,600 possible ways for the first three finishers to come in out of 50 runners when there are no ties. This is calculated using permutations.
Step-by-step explanation:
In a race with 50 runners where there are no ties, the number of ways the first three finishers can come in is calculated using permutations since the order of finishers matters. In this case, we want to find out how many different ways you can arrange the first three finishers out of 50 runners.
The formula for permutations is given by P(n, r) = n! / (n - r)!, where n is the total number of items to choose from, r is the number of items to arrange, "!" denotes factorial, and P(n, r) is the number of permutations.
For our specific case, where n is 50 and r is 3, the permutation is calculated as follows:
P(50, 3) = 50! / (50 - 3)! = 50! / 47! = 50 × 49 × 48 = 117,600.
Therefore, there are 117,600 possible ways for the first three finishers to come in.
Using the multiplication principle, we multiply the number of options for each place:
- Number of options for first place: 50
- Number of options for second place: 49
- Number of options for third place: 48
To find the total number of ways the first three finishers can come in, we multiply these numbers:
Total number of ways = 50 x 49 x 48 = 117,600