Three out of nine students in the computer club are getting prizes for first, second, and third place in a competition. How many ways can first, second, and third place be assigned?
Answer (2)
Recognize this as a permutation problem because the order of prize assignment matters.
Apply the permutation formula: P ( n , k ) = ( n − k )! n ! with n = 9 and k = 3 .
Calculate P ( 9 , 3 ) = 6 ! 9 ! = 9 × 8 × 7 .
Compute the final result: 504 .
Explanation
Understand the problem We have 9 students in the computer club, and we want to award prizes for first, second, and third place. Since the order matters (first place is different from second place, etc.), this is a permutation problem. We need to find the number of ways to arrange 3 students out of 9 in a specific order.
Apply the permutation formula The formula for permutations is given by: P ( n , k ) = ( n − k )! n ! where n is the total number of items, and k is the number of items we want to arrange. In this case, n = 9 (total number of students) and k = 3 (number of prizes).
Calculate the permutation Plugging in the values, we get: P ( 9 , 3 ) = ( 9 − 3 )! 9 ! = 6 ! 9 ! This means we have: P ( 9 , 3 ) = 6 × 5 × 4 × 3 × 2 × 1 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 We can cancel out the 6 ! from the numerator and denominator:
Find the final answer P ( 9 , 3 ) = 9 × 8 × 7 Now, we just multiply these numbers together: 9 × 8 × 7 = 72 × 7 = 504 So, there are 504 ways to assign first, second, and third place prizes to 3 out of 9 students.
State the final answer Therefore, the number of ways to assign first, second, and third place is 504.
Examples
Consider a scenario where a teacher wants to assign different roles (e.g., leader, secretary, and treasurer) to three students out of a class of nine. This problem is analogous to assigning prizes, as the order of selection matters. The teacher needs to determine how many different ways these roles can be assigned. Understanding permutations helps in such real-world scenarios where order and arrangement are important.
There are 504 ways to assign first, second, and third place prizes among 9 students in a computer club. This is calculated using the permutation formula, as the order of winning matters. By applying the formula and simplifying, we find the total arrangements.
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