Seven runners are competing in a race where 3 of them They will win medals for coming in first, second and third place. How many unique ways are there to order 3 of the 7 runners first? second and third place?
There are 7 choices for the first place finisher, then only 6 runners remaining for the second place finisher, and finally 5 runners remaining for the third place finisher. Therefore, there are $7 \times 6 \times 5 = \boxed{210}$ unique ways to order 3 of the 7 runners in first, second, and third place.
To find the number of unique ways to order 3 of the 7 runners for first, second, and third place, we can use the concept of permutations. The number of permutations of n objects taken r at a time is represented by nPr, which is calculated using the formula:
nPr = n! / (n - r)!
In this case, we have 7 runners and we want to order 3 of them, so we have:
n = 7 (number of runners)
r = 3 (number of medals)
Plugging these values into the formula, we get:
7P3 = 7! / (7 - 3)!
= 7! / 4!
Calculating further, we have:
7! = 7 * 6 * 5 * 4!
= 7 * 6 * 5 * 4 * 3 * 2 * 1
And:
4! = 4 * 3 * 2 * 1
Substituting these values back into the formula, we have:
7P3 = (7 * 6 * 5 * 4 * 3 * 2 * 1) / (4 * 3 * 2 * 1)
Simplifying the expression:
7P3 = 7 * 6 * 5
= 210
Therefore, there are 210 unique ways to order 3 of the 7 runners for first, second, and third place.
To find the number of unique ways to order 3 of the 7 runners for first, second, and third place, we can use the concept of permutations. A permutation calculates the number of possible arrangements of a set of items.
In this case, we need to find the number of permutations of 7 items taken 3 at a time because there are 7 runners and we want to choose 3 of them for first, second, and third place.
The formula for permutation is given by: P(n, r) = n! / (n - r)!
Where n is the total number of items and r is the number of items we want to choose.
In this case, n = 7 (number of runners) and r = 3 (number of places to assign).
Plugging the values into the formula, we get:
P(7, 3) = 7! / (7 - 3)!
= 7! / 4!
Calculating the factorials:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
4! = 4 × 3 × 2 × 1
Simplifying the expression:
P(7, 3) = (7 × 6 × 5 × 4 × 3 × 2 × 1) / (4 × 3 × 2 × 1)
= 7 × 6 × 5
= 210
Therefore, there are 210 unique ways to order 3 of the 7 runners for first, second, and third place.