Problem 57
Question
In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?
Step-by-Step Solution
Verified Answer
The first four finishers can come in 360 different ways in a six-automobile race.
1Step 1: Understand Permutation Formula
Permutation of n items taken r at a time can be calculated using the following formula: \(P(n, r) = n! / (n-r)!\). In our problem, n is 6 (as there are 6 automobiles) and r is 4 (as we need to find out the number of ways the first four automobiles can finish).
2Step 2: Substitute in the Permutation Formula
Put the values of n and r in the permutation formula: \(P(6, 4) = 6! / (6-4)!\). Now calculate the factorial of 6 and 2.
3Step 3: Calculate Factorials
Factorial of a number is the product of all positive integers up to that number. Factorial 6 is calculated as: 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 and factorial 2 is calculated as: 2! = 2 * 1 = 2.
4Step 4: Solve the Permutation
Replace the factorials with their results in the permutation formula (720/2) to find the number of ways.
5Step 5: Final Calculation
Perform the division to find out the final answer of the permutation. 720/2 = 360
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