Answer :
Final answer:
There are 42 ways to award the medals at the competition. (Option A)
Explanation:
Given:
Number of runners: 7
Number of medals to be awarded: 2
Calculate the number of permutations:
To determine the number of ways to award the medals, we use the concept of permutations. Permutations represent the number of ways to arrange objects in a specific order.
We use the formula for permutations: Pₙᵣ = n! / (n - r)!, where n is the total number of objects and r is the number of objects being arranged.
Substitute the given values:
For this problem, n = 7 (number of runners) and r = 2 (number of medals to be awarded).
So, we plug these values into the permutation formula: P₇₂ = 7! / (7 - 2)!.
Calculate the factorial:
Calculate the factorials in the formula. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 and (7 - 2)! = 5! = 5 × 4 × 3 × 2 × 1.
Perform the division:
Divide 7! by (7 - 2)! to get the number of permutations of 7 objects taken 2 at a time.
Calculate the result:
P₇₂ = 7! / (7 - 2)! = (7 × 6 × 5 × 4 × 3 × 2 × 1) / (5 × 4 × 3 × 2 × 1) = 7 × 6 = 42.
Therefore, there are 42 ways to award the medals at the competition, as determined by the permutation of 7 runners for 2 distinct medals. (Option A)